The Intertemporal Keynesian Cross

Adrien Auclert Matthew Rognlie Ludwig Straub

February 14, 2018

PRELIMINARYAND INCOMPLETE,PLEASEDONOT CIRCULATE

Abstract This paper develops a novel approach to analyzing the transmission of shocks and policies in many existing macroeconomic models with nominal rigidities. Our approach is centered around a network representation of agents’ spending patterns: nodes are goods markets at different times, and flows between nodes are agents’ marginal propensities to spend income earned in one node on another one. Since, in general equilibrium, one agent’s spending is another agent’s income, equilibrium demand in each node is described by a recursive equation with a special structure, which we call the intertemporal Keynesian cross (IKC). Each solution to the IKC corresponds to an equilibrium of the model, and the direction of indeterminacy is given by the network’s eigenvector centrality measure. We use results from Markov chain potential theory to tightly characterize all solutions. In particular, we derive (a) a generalized Taylor principle to ensure bounded equilibrium determinacy; (b) how most shocks do not affect the net present value of aggregate spending in partial equilibrium and nevertheless do so in general equilibrium; (c) when heterogeneity matters for the aggregate effect of monetary and fiscal policy. We demonstrate the power of our approach in the context of a quantitative Bewley-Huggett-Aiyagari economy for fiscal and monetary policy.

1 Introduction

One of the most important questions in macroeconomics is that of how shocks are propagated and ampliﬁed in general equilibrium. Recently, shocks that have received particular attention are those that affect households in heterogeneous ways—such as tax rebates, changes in monetary policy, falls in house prices, credit crunches, or increases in economic uncertainty or income inequality. New empirical evidence has been brought to bear, and new decision-theoretic models have been developed, to quantify and understand the causal effect of these shocks on aggregate consumption.1 Yet both cross-sectional empirical evidence and decision-theory models deliver

1For empirical evidence using cross-sectional variation, see Johnson, Parker and Souleles(2006) for tax rebates, on or Fagereng, Holm and Natvik(2016) on unanticipated income shocks. For mostly partial equilibrium decision-theory

1 partial equilibrium estimates of aggregate effects. A significant question that remains is how partial equilibrium effects translate into general equilibrium outcomes, once markets clear and the endogenous responses of monetary and fiscal policy are taken into account. This is the question we study in this paper. It is intuitive that there should be commonalities in the partial-to-general equilibrium adjustment mechanism. In a closed economy, whether consumption falls because of a fall in house prices, a credit crunch, or a fiscal retrenchment, common mechanisms are likely to operate to restore equality between equilibrium demand and output. One important such adjustment mechanism, highlighted by Keynes(1936), is the feedback between consumption and incomes: as consumption falls, incomes across the population falls, triggering further adjustments in consumption. Modern models used for monetary and fiscal policy analysis tend to emphasize other mechanisms instead: those that arise from the endogenous reaction of policy to the fall in economic activity. These models typically feature representative agents who smooth their consumption responses to income shocks over their infinite lifetime, rendering the consumption-income feedback quantitatively irrelevant. But a newer vintage of models incorporates much larger marginal propensities to consume out of transitory income shocks, which empirical evidence has consis- tently shown to be an important feature of the data. In these models, consumption-income feedbacks have the potential to play a much more important role for general equilibrium adjustment. We show that, in a broad class of models with nominal rigidities, the question of how to go from fundamental shocks to equilibrium outcomes is answered by the solution to one simple recursive equation, dY=∂Y+M dY (1) · In (1), the vector dY represents the general equilibrium impulse response to the shock–the change in the time path of output induced by the shock, which is the ultimate object of interest to macroeconomic analysis. ∂Y represents the partial equilibrium impulse response, whose definition we establish precisely below, and which in our benchmark case corresponds to the effect of the shock holding interest rates and quantities constant, as in the aforementioned literature. Finally, M is a matrix of general equilibrium feedbacks, encapsulating the response of private individuals to changes in incomes and interest rates, as well as endogenous policy responses, notably that of monetary policy. In a benchmark case in which real interest rates are held constant by monetary policy (’constant-r’), the entries of the M matrix correspond to households’ average marginal propensities to consume, at each date, for income received at other dates, weighted by the incidence of income across the population. This renders the M matrix potentially observable directly, just as ∂Y can be, opening the promise of conducting general equilibrium analysis using a sufficient statistic approach under minimal model assumptions. Keynes(1936) proposed an answer similar to our equation (1), focusing exclusively on the income-consumption feedback mechanism and the immediate response of output to shocks. In models, see Auclert(2017) on the effects of monetary policy, Kaplan and Violante(2014) on tax rebates, and Berger, Guerrieri, Lorenzoni and Vavra(2015) on house prices.

2 his formulation, dY = ∂Y + MPC dY (2) · where ∂Y is the impulse to aggregate demand, dY is the equilibrium change in output, and MPC is the aggregate marginal propensity to consume out of income. Equation (2) can be solved to obtain dY = (1 MPC) 1∂Y, where (1 MPC) 1 = 1 + MPC + MPC2 + . . . is called the multiplier − − − − and reflects the accumulated consumption feedback amplifying the original impulse. Although this traditional ‘Keynesian cross’ embodies a useful intuition, it has a number of weaknesses. It is static, not dynamic. It does not require that budget constraints be satisfied: an impulse to demand from government spending comes out of thin air, rather than being offset by taxation or a cut in spending at some other date. It does not directly correspond to any standard, microfounded model. It does not give a role to monetary policy. It assumes that the response of the economy is determinate—an assumption modern monetary theory has shown is only guaranteed under particular types of policy rules. We show that, nevertheless, going back to these intuitions is useful to understand modern business cycle theory, and furthermore the parallel delivers novel results that had escaped the literature until now. First, when monetary policy maintains the real interest rate constant, the M matrix in (1) corresponds to a weighted average of marginal propensities to consume. This captures the static consumption-income feedbacks emphasized by Keynes, as well as richer intertemporal interactions: for instance, some income earned in period 1 will be spent in period 2, from which some income will be spent again in period 1. Budget constraints also impose a number of important modifications to the static view. First, it is not generally possible to simply solve for dY in (1) without further considerations. The crucial observation here is that all income is eventually consumed in net present value terms–the dynamic M matrix has an eigenvalue of 1, making inversion of I M impossible in general. On the other hand, partial equilibrium shocks − on their own do not create present value. In our dynamic case, we show that equation (1) can be solved, but that it generally has multiple solutions.2 This indeterminacy is inherent to models with nominal rigidities, and for the first time, we are sharply able to characterize it, by showing how it relates to properties of the M matrix. To understand the amplification mechanisms inherent to our model is useful to think of time periods as nodes of a network, as in figure1. Each new unit of income generated at a given node is spent, partly on itself and partly on every other node, according to relationships given by the matrix M. This round of spending generates additional income at each node, which is again spent according to the same pattern, and so on. The final outcome for the distribution of income across nodes is our main object of interest. It is the general equilibrium effect on consumption after the intertemporal Keynesian cross has run its course. If the matrix M is written in net present value units—as we generally do—then the fact that all income is spent in net present value terms means that M is a left-stochastic matrix, with columns summing to one. It can then be interpreted as

2Intutitively, in the scalar version (2), budget constraints impose that ∂Y = 0 and MPC = 1, so any value of dY is a solution.

3 M ,0 · M0,0 M , · · M1,0 M1, ·

t = 0 t = 1 ···

M0,1 M ,1 · M1,1

M0, ·

Figure 1: Interpretating the economy as a network the transition matrix for a Markov chain, with dimension equal to the number of periods of the model. The stationary distribution of this Markov chain, if it exists, delivers the dimension of indeterminacy: if agents expect aggregate income to increase in proportion to this distribution over time, then their collective consumption responses are such that it sustain exactly this additional increase in income. Sometimes, however, when = ∞, the only stationary ‘distributions’ diverge T with time—an outcome that is natural to rule out, as the literature typically does. In this case, equilibrium is uniquely pinned down. We first show that, under constant-r monetary policy, whether or not the equilibrium is determinate depends on the economy’s M matrix. Specifically, to the characteristic function of the limiting distribution implied by the final columns of the M matrix—the time path of the response of consumption to an income shock far into the future. Intuitively, if income increases at a point in time tend to be collectively spent by agents later on—say because agents tend to wait to receive income before spending it due to borrowing constraints—then the economy is naturally determinate because only explosive patterns of income would self-sustain. Relaxing the assumption of constant real interest rates, we establish a New Taylor principle for equilibrium determinacy. When monetary policy endogenously raises interest rates in response to increases in economic activity (directly or indirectly through the effect of activity on inflation), agents collectively postpone their spending decisions, placing the economy in a determinate case. The Taylor principle characterizes exactly by how much nominal interest rates need to increase. The answer depends on the natural tendency of the economy to delay consumption relative to income, and the elasticity of the timing of consumption to changes in real interest rates. To the best of our knowledge, this is the first general Taylor principle ever derived in a heterogeneous agent setting. Our result also provides a new intuition for why the Taylor principle is necessary to ensure bounded equilibrium determinacy in standard New Keynesian models. We next show conditions under which partial equilibrium impulse responses can end up having non-zero net present value aggregate output implications, and relate it to the benefit of frontloading macroeconomic policies. The intuition there is that, when an economy is determinate, a policy that initially has a positive effect on aggregate demand but has a negative effect later

4 on–such as a ‘cash for clunkers’ program of temporary subsidies to car purchases–might nevertheless be a net positive because the later bust will be tempered by the rise in income occurring in the earlier period, boosting spending later on and offsetting the negative impulse. Finally, by exhibiting two benchmarks for fiscal and monetary policy, we show that our results allow us to quantify precisely when heterogeneity matters for the aggregate effect of policy, shedding new light on an important question in the literature. We illustrate all of our results in the context of an heterogeneous agent New Keynesian model, belonging to the new state of the art vintage of models used for policy analysis that are consistent with marginal propensities to consume observed in the data. We first show that the economy is determinate under constant real interest rates, provided income risk is not too countercyclical. If income risk becomes very countercyclical, however, agents expecting future falls in income start reducing their consumption today, an outcome that can become self-fulfilling. This relates our finding to those of Ravn and Sterk(2017), among others, who have shown the possibility of self- fulfilling increases in unemployment in a model with explicit search and matching. We next show that the Taylor principle is weakened in our baseline environment, relative to the representative agent case, and relate this to the amount of liquidity issued by the government. In the limit case of perfect liquidity, we recover the representative agent benchmark of a responsiveness of 1, while the economy can be determinate even under constant nominal interest rates in the limit as liquidity vanishes. We then study the magnitude of government spending multipliers, a topic that has received tremendous attention in the literature. We first establish a benchmark result for a balanced budget multiplier of 1 under the constant-r case, under a specific but natural assumption about tax adjustment. This generalizes Woodford(2011)’s landmark result to the case with heterogeneous agents, providing a useful benchmark for the quantitative literature going forward (see for example Hagedorn, Manovskii and Mitman(2017)). We show that, relative to this benchmark, multipliers are increased when the government delays levying the taxes. We relate this to our frontloading result: delaying taxes implies an additional fiscal stimulus that translates into a positive equilibrium output effect, and show that this effect can be quantitatively large–a stark departure from ricardian equivalence. Finally, we study monetary policy, comparing our results to the benchmark ‘as if’ case of Werning(2015). Contemporaneous monetary policy shocks are amplified relative to the representative agent case. This is almost entirely due to the income effect on consumption–those who gains from current fall in interest rates have higher marginal propensities to consume than those who lose—validating the result in Auclert(2017). Finally, we study forward guidance, and show that the effects are attenuated. This validates the result in McKay, Nakamura and Steinsson(2016), but for a different reason: those who gains from future falls in interest rates tend to be constrained and cannot increase their consumption today in anticipation of lower future payments, while those who lose are less constrained and therefore more responsive.

Related literature. Our paper relates to several strands of the literature. First, it is related to a recent literature on Heterogeneous Agent New Keynesian Models, which has studied, inter alia,

5 monetary policy and forward guidance (McKay and Reis(2016), Werning(2015), Kaplan, Moll and Violante(2016)), fiscal multipliers (Hagedorn et al.(2017)), or the role of precautionary savings in amplifying fluctuations (Ravn and Sterk(2013), den Hann, Rendahl and Riegler(2015), Bayer, Lütticke, Pham-Dao and von Tjaden(2015), Challe, Matheron, Ragot and Rubio-Ramirez (2014), Heathcote and Perri(2016)). We show that the that the M matrix encapsulates all that is needed to compare across these models. Second, our determinacy results relate to a long literature that has studied determinacy in New Keynesian models. In standard representative agent results (Woodford(2003), Bullard and Mitra(2002)), the Taylor principle is independent of features of the economic environment, except the slope of the Phillips curve and the discount factor of households. Recently, limited forms of Taylor principles have been obtained for economies with richer heterogeneity, eg in Galí, López-Salido and Vallés(2007), Bilbiie(2008), Coibion and Gorodnichenko(2011), or Ravn and Sterk(2017), the results depend on deeper features of the economy environment.Third, we related to a literature on sufficient statistics for heterogeneous agent models (Auclert(2017), Berger et al.(2015), Auclert and Rognlie(2016)). These papers have obtained sufficient statistics for partial equilibrium effects (∂Y). Our key innovation here is to pave the way for taking this methodology to the general equilibrium, by obtaining an M that is theoretically observable under the constant-real rate policy. Finally, our network related results relate to a literature on network theory and business cycles, in the tradition of Long and Plosser (1983), or more recently Acemoglu, Carvalho, Ozdaglar and Tahbaz-Salehi(2012). An important distinction between this literature, which models production networks with intermediate inputs, and our results is that they work with some fixed factor of production rendering the Leontieff matrix I M invertible, whereas in our case it is generally not invertible. Instead, we make heavy − use of the theory of infinite dimensional Markov chains, as covered in Kemeny, Snell and Knapp (1976), to obtain solutions to our leading equation.

2 Baseline model

In this section, we introduce a benchmark Heterogeneous Agent New Keynesian model that we will use for our numerical illustrations throughout the paper. This is one of many models that can be characterized by an Intertemporal Keynesian Cross, as we discuss further in the next section.

Model setup Time is discrete and runs from t = 0 to , with the inﬁnite horizon case = ∞ T T having special interest. The economy is populated by a continuum measure 1 of ex-ante identical agents who face no aggregate uncertainty, but may face idiosyncratic uncertainty. Individuals can be in various idiosyncratic ability states ei, and transition between those states according to a Markov process with ﬁxed transition matrix Π. We assume that the mass of worker type i in idiosycratic state ei is always equal to π (ei), the probability of ei in the stationary distribution of E = = Π. Ability levels are normalized to be one on average: e [e] ∑ei π (ei) ei 1.

6 Agents. Agents have time-0 utility over consumption and labor supply plans given by separable preferences " # t E ∑ β ut (cit) v (nit) (3) t 0 { − } ≥ Each period t, an agent with incoming real wealth ait 1 and realized earnings ability eit enjoys − the consumption of a generic consumption good cit and gets disutility from working nit hours.

Consumption goods have price pt and the nominal wage per unit of ability is wt. The agent receives a lump sum transfer from the government tt, pays a marginal tax rate τt on labor income, can trade in real bonds that deliver a net real interest rate rt, and faces a borrowing constraint. Speciﬁcally, his budget constraint in period t in units of date-t consumption goods is

wt cit + ait = (1 + rt) ait 1 + tt + (1 τt) eitnit t, i (4) − − pt ∀ a a (5) it ≥ t

The agent maximizes (3) by choice of cit and ait, subject to (4), the borrowing constraint (5), and, when < ∞, a terminal condition ai = 0. By contrast, due to frictions in the labor market, T T the agent is restricted to supply nit hours. Hence the agent takes total net of tax income zit = wt tt + (1 τt) eitnit as given. From the household’s perspective, therefore, equation (4) simpliﬁes − pt to

cit + ait = (1 + rt) ait 1 + zit t, i (4’) − ∀ Maximization of (3) subject to (4’) and (5) constitutes the core of the standard incomplete market model of consumption and savings.

Labor market. Following standard practice in the New Keynesian sticky-wage literature, labor hours nit are determined by aggregate union labor demand: when employment is lt, households must supply

nit = Γ (eit, lt)

As in Werning(2015) and Auclert and Rognlie(2016), the Γ function—which we refer to as the gross incidence function—allocates hours among heterogeneous households as a function of aggregate employment. That function respects aggregation, in that skill-weighted hours worked are always equal to aggregate labor demand

E [eΓ (e, l)] = l l e ∀

Households are off their labor supply curves because the nominal wage wt is partially sticky. In appendix we formalize this wage stickiness by assuming that that there is a continuum of unions j [0, 1], each providing a union-speciﬁc task l and employing each of the workers, and each only ∈ jt occasionally resetting their wage in a Calvo fashion. In a steady-state with constant employment

7 l, this formulation implies that the real wage w∗ is at a markup over some average across all p∗ households of the marginal rate of substitution between consumption and hours. Outside of steady-state, we make the assumption that wage resets take place under the assumption that the consumption distribution remains at its steady-state level. We then show that this problem leads to a simple New Keynesian Phillips curve for aggregate wage growth, πw log Wt , given to ﬁrst order by t Wt 1 ≡ − 1 πw = λ lb (w p ) + β πw (6) t t ψ t − t − t t t+1 where l log (l /l), ψ is a constant, and κ , β are deterministic functions of time, with β = β t ≡ t t t t when = ∞. T Our assumption that the consumption distribution remains at its steady-state level when wages are reset implies that, outside of steady state, wealth effects on labor supply are absent. This is important to obtain some of our results mapping partial to general equilibrium effects—a point we return to further in section 3.6.

Production of ﬁnal goods. Firms operate a simple, time-invariant technology

yt = lt (7)

They are perfectly competitive and set prices ﬂexibly, so that the ﬁnal goods price is given by

pt = wt (8) and profits are 0, justifying why no dividends enter households’ budget constraints in (4). Pt Hence, goods price inflation πt log and aggregate wage inflation are identical at all ≡ Pt 1 w − times, πt = πt , and the Philips curve (6) can also be written as

πt = κtybt + βtπt+1 (9) where κ = λt and y log (y /y), where y in steady state output. t ψ bt ≡ t

Government. In each period, the government has rule for its outstanding amount of real debt bt = bt, and spending spending gt = gt. It then adjusts a combination of the lump-sum tt and the marginal tax rate τt so as to satisfy its budget constraint

wt τt lt tt = (1 + rt) bt 1 + gt bt revt (10) pt − − − ≡

8 where revt represents net government revenue. Speciﬁcally, we assume that there exists a sequence r τt and a constant ϕ such that tax revenue and the lump-sum rebate are, respectively,

wt r τt lt = τt yt + (1 ϕ) revt (11) pt −

t = τry ϕ rev (12) t t t − · t At time t, τr [0, 1] corresponds to the share of GDP that the government would use for transfers t ∈ to households if it did not require revenue to pay for spending or interest on the debt. In addition, ϕ [0, 1] speciﬁes how the government adjusts taxes vs transfers at the margin when it requires ∈ an extra unit of revenue. ϕ = 0 correponds to the case where it only raises taxes, ϕ = 1 where r it only lowers transfers. When τt = ϕ, all changes to revt affect households’ net-of-tax incomes wt tt + (1 τt) eitnit in proportion, irrespective of their skill level eit. − pt Monetary policy sets the nominal interest rate it by following one of two rules described below.

Given the path for good prices pt, the real interest rate at t (the price of date-t + 1 goods in units of date-t goods) is then equal to pt 1 + rt (1 + it) (13) ≡ pt+1 The ﬁrst rule we consider is a constant rule for the real interest rate

rt = rt (Constant-r)

Such a rule has proved useful in the equilibrium analysis of both representative- and heterogeneous- agent New Keynesian models,3 and we will show that it facilitates simple results in our context as well. We will also consider the consequences of assuming that monetary policy follows a conven- tional Taylor rule it = it + φππt + φyybt (Taylor rule) where, at this stage, φπ and φy are unrestricted, and we provide conditions for equilibrium determinacy in section 5.2. Definition 1. Given exogenous sequences for preferences u , borrowing constraints a , fiscal n o t t r { } policy bt, gt, τt , and monetary policy rt, it , a general equilibrium in this economy is a path for prices p , w , r , i , aggregates y , l , c , b , g , t , individual allocations rules c (a, e) , n , and { t t t t} { t t t t t t} { t it} joint distributions over assets and productivity levels Ψ (a, e) , such that households optimize, { t } unions optimize, firms optimize, monetary and fiscal policy follow their rules, and the goods and bond markets clear: Z gt + ct (a, e) dΨt (a, e) = yt (14) Z adΨt (a, e) = bt (15)

3See for example Woodford(2011) or McKay et al.(2016),

9 Parameters Description Value ν Elasticity of intertemporal substitution 0.5 β Discount factor 0.962 r Real interest rate 4% π Inﬂation rate 0% b/y Government debt to GDP 140% a Borrowing constraint 0 g/y Government spending to GDP 20% τr Tax rate for lump-sum 17.5% τ Marginal tax rate 38% κ Slope of the Phillips Curve 0.1

Table 1: Calibrated parameters

This completes the description of our environment. The model cannot be solved analytically and one has to resort to numerical simulations for solutions. However, in the next section, we show that we can make conceptual progress in characterizing impulse responses from steady state.

Calibration. For our numerical simulations, we stay as close as possible to a standard calibration for a Huggett model. We consider the model in infinite horizon ( = ∞), and assume that 1 ν 1T c − − households have constant CES utility over consumption u (c) = 1 with a standard value of 1 ν− 1 − ν = 2 . We set a steady-state value for output of y = 1, a steady-state inflation rate of π = 0, and assume a slope for the Phillips curve of κ = 0.1. Conditional on these choices, the specific form of labor disutility v (n) and the price rigidity parameter θ are unimportant. We follow the recent incomplet markets literature in calibrating the gross income process to features of earnings changes from W2 data, and the tax function to features of the US tax-and- transfer system. In this spirit, our gross income process is the same as in Kaplan et al.(2016), while we follow Auclert and Rognlie(2016) and assume that the lump-sum transfer component of taxation represents τr = 17.5% of GDP. We assume that all individual hours change proportion to aggregate employment, Γ (e, l) = l, and set the marginal tax adjustment parameter to ϕ = τr. Together, these two assumptions imply that changes in aggregate income lt affect the net-of-tax incomes of all households in proportion. We refer to this as the constant incidence case. Werning(2015) established this case as a benchmark for the analysis of monetary policy, and we will show that it is a natural benchmark for the analysis of fiscal policy as well. We follow McKay et al.(2016) and assume that households cannot borrow ( a = 0) and that b government debt is y =140% of output—an assumption meant to capture the amount of liquid assets in the US economy more broadly. We target an equilibrium real interest rate of r = 4%, g and assume that government spending is y = 20% of output. Together, these assumptions r r b g imply an overall marginal tax rate of τ = τ + (1 τ ) r y + y = 38% and a lump sum of − t = τr 1 r b + g = 13%. Finally, we choose the discount factor β so as to achieve our target y − y y

10 real interest rate in the economy at steady state. Table1 summarizes our calibration.

3 Deriving the Intertemporal Keynesian Cross

We assume that the economy starts in a given steady state with r > 0. We index all sequences that are exogenous in our definition of general equilibrium by a single parameter e. Our objective is to understand the equilibrium response of the economy, starting from steady state, to a small change in this e. This encompasses a broad class of shocks that these models enable us to study: shocks r to preferences (ut), borrowing constraints (at), fiscal policy (bt, gt, τt ) and monetary policy (rt, it). Note further that this formalism captures both unexpected shocks at date 0 (for example r0 > 0, a contemporaneous monetary policy shock) as well as expected shocks that agents anticipate ahead of time (for example rT > 0 for T > 0, a forward guidance shock). In a setup such as ours, the shocks we consider all have complex general equilibrium effects. This complexity mainly stems from intertemporal feedbacks: a shock to income in the future rever- berates, via agents’ consumption smoothing decisions, to the entire path of consumption. Changes in real interest rates—as induced by endogenous inflation or responses of monetary policy—have a similar feature: they elicit complex patterns of intertemporal substitution responses, as well as income effects that are both difficult to characterize theoretically and hard to map to data. When these feedback mechanisms are strong enough, the model may even feature multiple equilibria. We break through this complexity by defining a simpler notion of equilibrium which we call the exogenous y equilibrium (or EYE) that shuts down these effects in response to the shocks we consider. We show that this definition nests a simple notion of partial equilibrium (PE) that the literature has considered. This PE effect is conceptually a lot simpler, and is an object that can conceivably be measured in data following the lead of a recent ‘sufficient statistic’ literature. Next, we consider how the general equilibrium (GE) effect relates to the PE effect. We show that, to first order, the differences between PE and GE are captured by a simple linear mapping summarizing all the intertemporal feedbacks described above. We characterize how the M matrix varies under different policy rules, and show that they relate to households’ marginal propensities to consume out of income and interest rates. This opens up the possibility of mapping the M matrix to observables as a key element of model validation. We finally briefly illustrate the usefulness of this approach for model comparison by showing how alternative models can all be summarized through the M matrix, before turning to the solution of the model in the next section.

3.1 The exogenous y equilibrium (EYE)

We start by deﬁning our notion of an exogenous y equilibrium.

Deﬁnition 2. Given a value of e, and a path for output supply y , an exogenous y equilib- { t} rium (EYE) is a path for prices p , w , r , i , aggregates y , l , c , b , g , t , individual allocations { t t t t} { t t t t t t}

11 Inﬂation π Output supply y and shock e t { t}

Monetary policy Household incomes Fiscal policy

Consumption c Government spending g { t} { t}

Output demand yEYE { t }

Figure 2: Uniqueness of EYE mapping yEYE ( y ) { t} rules c (a, e) , n , and joint distributions over assets and productivity levels Ψ (a, e) , such { t it} { t } that households optimize, unions optimize, firms optimize, and monetary and fiscal policy follow their rules. We define yEYE c + g as the resulting path of output demand. t ≡ t t

Deﬁnition2 drops the goods market clearing condition yt = ct + gt and instead assumes that the path of yt is given. This shuts off most of the complex intertemporal feedbacks in the model, since there is now a simple directional ﬂow from a path of output supply yt to a path of ag- EYE { } gregate output demand yt . The following lemma shows formally that there exists such a mapping.

EYE Lemma 1. There is a unique path for output demand yt corresponding to a given path for output supply y and shock e. This defines a mapping yEYE ( y ; e). { t} t { t} The proof of lemma1 is illustrated in figure2. Given e and a path for output supply, firm optimization implies that employment is lt = yt, by (7). Given this, union optimization implies an updated path for aggregate nominal wages w and individual employment n . In turn, { t} { it} final good firm optimization determines the path for prices p = w , and therfore inflation π . t t { t} Next, given y , π and the shock sequence, we obtain the paths for real interest rates r and { t t} { t} the taxes t , τ that households face. Together, these deliver all of the elements that enter as an { t t} input into households’ decision problem: a state-by-state path for aftertax incomes and a path for real interest rates. Households optimally choose their consumption plans given these paths, and aggregating those decisions using the wealth distribution at every point we obtain the partial equilibrium path c , while using the fiscal rule we obtain the path for government spending { t} g . Summing the two, we obtain output demand yEYE. { t} t

12 The only difference between deﬁnitions1 and2 is the requirement of goods market clearing. We therefore obtain the following lemma.

Lemma 2. A general equilibrium is an exogenous y equilibrium that satisﬁes the additional requirement that goods market clear at every point in time, that is, a path y that solves the system of equations { t}

y = yEYE ( y ; e) s (16) s s { t} ∀

When (16) holds for all t, it follows from Walras’s law that (15) holds for all t as well. A key object for our analysis will be that of an EYE that has constant output equal to the steady state value y = y . We deﬁne the sequence ∂y as { t} { } { t} ∂yEYE ( y , e) ∂y t { } de (17) t ≡ ∂e In the next section, we relate this to the partial equilibrium household problem. In the following subsection we then show the sense in which ∂yt is sufﬁcient for the general equilibrium effect.

3.2 Partial equilibrium interpretation

A large literature has studied partial equilibrium household problems. We adapt this standard deﬁnition to our context below. Deﬁnition 3. Given exogenous sequences for preferences u , borrowing constraints a , real { t} t interest rates r and a stochastic process for net incomes z (e) , a partial equilibrium for house- { t} { t } holds is a set of individual allocations c (a, e) and joint distributions over assets and productivity { t } levels Ψt (a, e) , such that households optimize, and the evolution of Ψt is consistent with house- { } R hold policies. We write aggregate consumption as cPE c (a, e) dΨ (a, e). t ≡ t t PE For each of our shocks, we can now describe ∂yt in terms of a particular ct response to a change in inputs.

Proposition 1. For each of the following shocks, starting from steady state, we have

PE ∂yt = dct + dgt

PE where dgt is the perturbation to government spending and dct is the ﬁrst order response of partial equilibrium consumption to the following changes: a) For preferences u or borrowing constraints a , to the shocks themselves { t} t n o r b) For ﬁscal policy τt , gt, bt , to a change in net labor income

dz = y (1 ω ) dτr ((1 ϕ) ω + ϕ) drev (18) it − it t − − it t

13 where ω eitnit = eitΓ(eit,y) is i’s relative gross income (with E [ω ] = 1), and drev = it ≡ y y I it t (1 + rt) dbt 1 + dgt dbt is the change in required government revenue induced by the new ﬁs- − − cal path c) For monetary policy it, rt , to a change in real interest rates drt = drt or drt = dit, and a change in

net labor income (18), where drevt = bt 1drt is the change in required government revenue induced − by the new path for interest rates

Proposition1 shows that, for shocks that affect the household problem directly, the EYE response ∂yt corresponds exactly to the traditional partial equilibrium consumption demand re- PE sponse dct . For ﬁscal shocks, EYE includes two components: the direct change in government spending dgt, if any, and the partial equilibrium consumption response to the change in taxes. For shocks to the monetary rule, EYE combines two effects on household partial equilibrium consumption: the effect of changing rt, and the effect of changing taxes due to the government’s changing interest obligations.

This reveals a practical advantage of EYE: calculating ∂yt requires only the ability to calculate partial equilibrium consumption responses, sidestepping the complexity of general equilibrium.

Since only such computations are involved, we choose to refer to ∂yt as the partial equilibrium effect of a shock. For a fiscal or monetary shock, this concepts includes the consumption impact of the tax changes required to balance the government budget. The major advantage of adopting this definition of a partial equilibrium effect is that ∂y will { t} turn out to be sufficient to determine the general equilibrium output effect of any shock—something that alternative definitions could not deliver. Under our definition, all partial equilibrium effects also share the following unifying property:

Lemma 3. All partial equilibrium effects ∂yt have zero present value, that is,

T ∂yt ∑ t = 0 (19) t=0 (1 + r)

The proof of lemma3, in appendix C.1.2, follows from the fact, in our deﬁnition of EYE, all agents respect their budget constraints. By our assumption of constant y, the shocks we consider do not on their own generate any additional income in the aggregate, even though usually are redistributing between agents. Therefore, they cannot lead to a change in the present value of aggregate spending, even though they can affect its path over time. Lemma 19 relates the popular idea that ﬁscal stimulus programs such as cash for clunkers “steal demand from the future”. With our notion of EYE this is true of all shocks, including redistributive shocks, government spending shocks and monetary policy shocks.

Note ﬁnally that ∂yt has a simple interpretation as the aggregate goods demand impulse to shock in a small open economy (interpreting changes in monetary policy as changes in the world

14 interest rate), provided the economy starts out in a steady state with a zero net international asset position.

Empirical moments for partial equilibrium effects. An important beneﬁt of the partial equilibrium concept is that, since all intertemporal effects have been shut down, it is either measurable directly (as in the case of a self-ﬁnancing government spending change), or corresponds to simple data moments. For example, Auclert and Rognlie(2017a) show that, for a redistributive tax rate r shock dτ0 taking place at date 0,

dτ0 ∂y = Cov (mpc , z ) r (20) t I i0t i0 1 τ0 − r where mpci0t is agent i’s average marginal propensity to consume at date t for shocks received at date 0, and zi0 is agent i’s net income at date 0. Conveptually (20) is measurable in data containing information on marginal propensities to consume and incomes. Relatedly, Auclert(2017) shows sufﬁcient statistics for the effect of a time-0 monetary policy shock, Guerrieri and Lorenzoni(2017) have one for the effect of a deleveraging shock, and Berger et al.(2015) for the effect of a house price shock.4 These empirical moments promise discipline for general equilibrium models, but share the property that they all correspond to partial equilibrium effects. We next show the sense in which partial equilibrium effects are sufﬁcient for the general equilibrium effect.

3.3 General equilibrium and the Intertemporal Keynesian Cross

Deﬁne yt (e) to be a function mapping each shock e to a general equilibrium. (If there are multiple equilibria for a given e, this can be any differentiable selection). Write dy dyt de. We can now t ≡ de relate dy to the partial equilibrium ∂y from the previous section. { t} { t} Totally differentiating (16), we obtain

T dys = ∂ys + ∑ ms,tdyt s (21) t=0 ∀ where EYE ∂ys ( y , e) ms,t { } ≡ ∂yt For any given ∂y , a general equilibrium dy is a solution to the linear system in (21). { s} { t} T × T We can prove the following analogue of lemma3:

Lemma 4. For any t, the present value of ms,t discounted to date t equals one:

T ms,t ∑ s t = 1 (22) s=0 (1 + r) − 4House prices shocks are outside of the scope of our model, but many papers study their effect on the aggregate demand for consumption without modeling the general equilibrium feedbacks between income and consumption.

15 As with lemma 21, this comes from the observation that all income earned in the economy is spent at some point in time. Incrementing yt by one therefore also increases the date-t present value of goods demand across all periods by one.

The Intertemporal Keynesian Cross. Our main result, the Intertemporal Keynesian Cross, is an expression of equation (21) in vector form. Since lemmas3 and4 will prove to be important, it is convenient to rescale quantities so that they have especially simple vector interpretations. We 1 1 1 therefore write dYt t dyt, ∂Yt t ∂yt, and Ms,t t s ms,t. All other capital letters ≡ (1+r) ≡ (1+r) ≡ (1+r) − will refer to present value concepts as well. Define the -dimensional vector ∂Y (∂Y , ∂Y , ) and the matrix M (M ), T ≡ 0 1 ··· T × T ≡ s,t recalling that can be infinite. The system of equations (21) delivers our main proposition. T Proposition 2 (The Intertemporal Keynesian Cross.). To first order, for any shock, impulses to partial equilibrium output ∂Y and general equilibrium output dY satisfy

dY = ∂Y + MdY (23)

The matrix M is column stochastic (it has a left eigenvector with eigenvalue 1, 10M = 10), and the vector ∂Y has mean zero, 10∂Y = 0.

We are interested in characterizing the solution(s) to equation (23). Lemma1 implies that for any shock, ∂Y is unique. Further, it implies that if dY is unique, all equilibrium objects locally also are. The question of local equilibrium muliplicity is therefore reduced to the question of whether, for a given ∂Y, there is a unique solution dY to equation (23). Characterizing when this is true is a major topic of sections4 and5 of this paper. For now, we observe:

Corollary 1. When equilibrium is locally unique, ∂Y is sufficient for dY: shocks that have the same partial equilibrium effect also have the same general equilibrium effect. Moreover, there exists a unique matrix , G invariant across shocks, such that dY = ∂Y G We call (23) the Intertemporal Keynesian Cross because it relates any impulse to aggregate demand ∂Y to its general equilibrium effect dY. The fact that there exists a simple mapping is very important: it implies that all shocks affect aggregate output through their partial equilibrium effect, irrespective of the source of that shock. Clearly, the matrix relates to M, even though G inverting I M is generally infeasible. − What is the benefit of this approach? It allows us to split conceptually the analysis of general equilibrium into two parts. First, how does the policy affect the economy in partial equilibrium? (∂Y) Second, how do partial equilibrium effects translate into general equilibrium effects? ( ). It G shows clearly why the underlying mechanisms of adjustment from partial to general equilibrium are common, once partial equilibrium is defined in the way that we do. This provides useful discipline, both for empirical work and for theoretical research going forward.

16 Network interpretation. Since M is a Markov chain, we can interpret equation (23) as an equation for ﬂows on a network

T dYs = ∂Ys + ∑ Ms,tdYt s (24) t=0 ∀

In this network, nodes are time periods, dYt are values of demand at each node, and Ms,t EYE ≡ ∂ys ( y ,e) s{ t} is the economy’s aggregate marginal propensity to spend out at date s out of aggre- (1+r) − ∂yt gate income at date t, which represents the strenght of flows across the network, with lemma 22 showing that all flows are conserved, ie, ∑sT=0 Ms,t = 1. Equation (24) then says that the value of demand at node s must be equal to the exogenous value ∂Ys plus the sum of entering flows from other periods. We will come back to this useful analogy when discussing the solution in section4.

3.4 Structure of the M matrix and relation to empirical MPCs

We now turn to the relationship between the M matrix and determinants of household behavior and policy.

The M matrix under a constant-r rule. We ﬁrst characterize the M matrix under the assumption that the monetary policy rule enforces an exogenous path for the real interest rate.

Lemma 5. Assume that monetary policy follows the rule (Constant-r). Then M = MY, where the MY matrix is deﬁned as Y EI [mpcist] Ms,t s t (25) ≡ (1 + r) −

E0[∂cis] where mpcist is i’s average marginal propensity to consume income received at date s. ≡ ∂yt The reason why the M matrix is so simple under the assumption of constant real interest rates is that the only way in which an aggregate income change affects aggregate spending is through the spending patterns of agents who receive that extra income. By contrast, if income changes generate endogenous changes in real interest rates, they will elicit aggregate income and substitution effects. We turn to this more complex case next. For our benchmark economy, we compute the MY matrix numerically, and plot it in ﬁgure3. Y Consider ﬁrst the term M0,0 is the upper left corner. According to (25), this term is Y ∂ci0 ∂ci0 ∂zi0 M0,0 = EI = EI (26) ∂y0 ∂zi0 ∂y0

Y where zi0 is the net income of agent i at date 0. Equation (26) shows that M0,0 is the average marginal propensity to consume out of individuals out of net income, ∂ci0 , weighted by the inci- ∂zi0 ∂zi0 dence of aggregate income for individual net income, γi0 . In our benchmark case of equal ≡ ∂y0 zi0 incidence of net incomes ni0 = l0 = y0, it is easy to show that γi0 = . The intuition is that EI [zi0] every additional dollar of GDP goes to the household sector: the government cuts the tax rate and

17 0.3 0 1 2 3 0.2 4 5 6 0.1

Date income spent 7 8 9 0.0 0 1 2 3 4 5 6 7 8 9 Date income received

Figure 3: Numerical M matrix for our HANK economy raises the lump-sum in such a way that all indidviduals are affected in proportion to their current Y income zi0. Hence, in that case, M0,0 is just the net-income-weighted average of MPCs. In our calibration that number is 0.25, consistent with the empirical evidence of how households spend tax rebates. The M matrix contains other terms, however. First, it tells us how a contemporaneous increase in GDP ends up being spent over time (the left column). Second, it tells us how future increases in GDP are spent. In this case, what matters is both future MPCs and expected incidence of a future increase in aggregate income. A salient feature of this matrix is that it is close to diagonal and that, especially after the ﬁrst few periods, columns tend to have the same structure shifted down along the diagonal. We say that the M matrix is asymptotically self-similar and will exploit this feature in section 5.1.

The M matrix under a Taylor rule. Under more standard rules for monetary policy such as (Taylor rule), the M matrix also reﬂects the way in which spending is shifted towards other dates to the endogenous reaction of real interest rates to aggregate income changes at a given point in time. Deﬁne the matrix Φ with element (s, t) given by ∂rs , capturing the effect that an increase in ∂ybt current income y has on the real interest rate in period s. Assume further that = ∞, so that the bt T Phillips curve (6) is simply πt = κybt + βπt+1 (27)

18 Combining (Taylor rule), (27), and the approximation it = rt + πt+1 to the Fisher equation (13), we see that the matrix Φ has elements  0 s > t  ∂rs  Φs,t = φy + κφπ s = t ≡ ∂yt b   φ β 1 βt sκ s < t π − − −

Output increases raise real interest rates contemporaneously by both the direct effect φy and the indirect effect due to inﬂation κφπ. Future output increases affect both inﬂation both today and tomorrow. The former effect leads to an increase in nominal interest rates due to the Taylor rule, while the latter leads to a decline in real interest rates via the Fisher equation. We can then use the chain rule to obtain:

Lemma 6. Assuming that monetary policy follows (Taylor rule), then

M = MY + MRΦ where 1 R E Ms,t s t I [mpcrist] ≡ (1 + r) −

E0[∂cis]/y and mpcrist is i’s average consumption response at date s for small changes in real interest rates ≡ ∂rt at date t.

The M matrix now captures both the direct household spending reaction, as well as the indirect spending reaction induced by a change in real interest rates via the endogenous monetary policy rule. Since the household sector intermediates the effect of changes in real interest rates on aggregate demand, its matrix of reactions to real interest rates MR now matters. Note that changes in real interest rates are just another type of partial equilibrium shock, and so all columns of MR sum to 0. Therefore, conservation of ﬂows is still satisﬁed.5

Using the M matrix for model validation. Lemmas6 and5 show the centrality of the matrix of marginal propensities to consume (contained in MY) in determining the general equilibrium M matrix. Together with ∂Y, for which sufﬁcient statistics already exist, the MY matrix is therefore a key determinant of equilibrium. While the literature already understands that it is important for aggregate models of ﬁscal and monetary policy to match average marginal propensities to Y consume as estimated in data (i.e., M0,0), these lemmas show that it is also important, in principle, to match data on how steeply these MPCs decay over time after agents receive transfers, as well as their anticipated spending ahead of transfers. To date, limited empirical evidence exists on such dynamic MPCs (see Broda and Parker(2014) for an example estimation following the 2008 tax rebates). Our theory shows why it is important

5 Y R Speciﬁcally, 10M = 10M + 10M Φ = 10 + 0

19 to gather more information on these, as they promise to discipline general equilibrium models going forward.6

3.5 The M matrix as a model comparison tool

Even though we have derived Proposition2 in the context of a speciﬁc model, it should be clear that a similar equation holds in any model in which a function yEYE can be deﬁned. In particular, we can easily relax assumptions about preferences as well as population structure—for example, introducing life-cycle considerations, overlapping generations, many different permanent types of agents, alternative asset market structures (agents that can insure their idiosyncratic shocks or hand to mouth agents that have no access to any asset).The main important ingredient is budet constraints, which holds in all models. This allows us to use the M matrix as a tool to compare across models.

Representative-agent economy. An inﬁnitely-lived representative agent economy in which β (1 + r) = 1 has MY matrix   m m m  ···   mβ mβ mβ   ···  Y  2 2 2  M =  mβ mβ mβ  (28)  3 3 3 ···   mβ mβ mβ   . . . ···.  . . . .. with m = 1 β is the marginal propensity to consume. This reﬂects the fact that the consumption − of each agent obeys the permanent income hypothesis, so that its marginal propensity to consume is the same in every period, no matter when the income is received.

OLG economy. In appendix B.1 we describe an OLG economy, inhabited by young and old agents at a given point in time with discount factor β. We show that, under constant-R monetary policy, the MY matrix for that economy is

  m 0    1 m m 0   −   ..  MY =  0 1 m . 0  (29)  −   .   . 0 m 0   . .  0 . 1 m .. − where m 1 is the marginal propensity to consume of a young generation. ≡ 1+β 6Our theory also suggests the importance of gathering information on the time path of household responses to changes in interest rates, which is important for the transmission mechanism through endogenous monetary policy.

20 Hand to mouth agents. An important part of the New Keynesian literature has argued that the presence of hand-to-mouth agents, who just consume their income in every period, signiﬁ- cantly alters the transmission mechanism of monetary and ﬁscal policy (see for example Galí et al. (2007) or Bilbiie(2008)). Consider an economy whose M matrix would be MYn without hand to mouth agents, and assume that there are a fraction µ of such agents, each with a unit elasticity to aggregate income Γ (e, l) = l. Then, the economy’s MY matrix is

MY = (1 µ) MYn + µI (30) −

For example, if the economy is a Two-Agent New Keynesian model (TANK) with permanent income and HtM agents, as in the literature cited above, then as the fraction of hand to mouth agents increase, the MY matrix moves away from the ’flat’ pattern in (28) toward a diagonal pattern. In- tuitively, this increases the amount of current-income amplification. Note, however, that the MY matrix depicted in figure3 also contains positive elements close to the diagonal. There is a significant debate in the literature as to whether heterogeneous agent models such as the one we write down can be correctly approximated by two-agent models of this kind. This result suggests exactly what the TANK approximation misses: feedbacks from periods around the shocks, that agents without access to financial markets are insensitive to, while agents with some ability to smoothe across periods are able to respond to.

3.6 Discussion

As we have stressed in this section, our approach has a number of distinct advantages over a standard computational approach, which would be to simply solve for general equilibrium directly. It gives us insights into the equilibrium response, illustrating the common adjustment mechanisms across shocks. The M matrix can be used for model validation as well as for model comparison. In the next section we will also be able to show that it helps characterize when the model has a unique equilibrium. On the other hand, this approach also comes at the cost of a number of assumptions. Some of these assumptions are common to general equilibrium models: agents satisfy budget constraints and interact via markets in which they fact the same prices. The most important limiting assumption we have made is the absence of wealth effects on labor supply: agents desired labor supply does not change as their income changes. This assumption had two parts: ﬁrst, wage rigidities impose that agents do not choose how much they work in the short run, and second, unions do not take the consumption distribution into account when resetting wages, eliminating medium- run wealth effects on labor supply as well. By contrast, our assumption of wage rigidities can be easily relaxed, provided we maintain absence of wealth effects on labor supply.7

7GHH preferences and ﬂexible prices would be an alternative, but this setup has problematic implications for aggregate multipliers (see Auclert and Rognlie(2017b)). We therefore prefer wage rigidites in this context.

21 4 Solving the IKC

Having derived the IKC in Proposition2 in the context of our HANK economy and having discussed its applicability to wide range of alternative models in Section 3.5, we can now turn to its solution. Section 4.1 provides an existence result and Section 4.2 establishes conditions that ensure (local) determinacy of the solution. To be widely applicable, this section only relies on the IKC (23) and not on any other aspects of the model. It is therefore essential to state all assumptions on the objects entering the IKC. ( +1) ( +1) We assume that M satisfies the assumptions of our column-stochastic matrix in R T × T , T+1 ∑T M , = 1, and that ∂Y R has zero net present value (NPV), that is, ∑T ∂Y = 0. In t=0 t s ∈ t=0 t order to be able to use tools from Markov chain theory, in this section, we further assume that M is non-negative M 0.8 We allow to be either finite or infinite, in which case any sum is well- t,s ≥ T defined whenever each the associated sequence of partial sums converges. Similarly, the product of two infinite-dimensional matrices is well-defined whenever each element, itself an infinite sum, is well-defined. All proofs of results in this section are contained in AppendixC.

4.1 General solution of the IKC

Two main complications arise when trying to solve the IKC (23) for dY. First, the IKC cannot be solved by simply inverting (1 M) to solve for dY since 1 M is generally not invertible: in many − − cases it has a right-eigenvalue of 1. For instance, when < ∞, this is an immediate consequence T of the stochasticity of M (Lemma4). This property of the IKC causes indeterminacy, which we will study in great detail below. Second, an equation of the form of the IKC (23) does not necessarily admit any solution, even with the assumptions in place so far. For example, suppose the MPC matrix M is the identity matrix. In that case, no solution dY exists whenever ∂Y = 0. Clearly, such an MPC matrix would not 6 be a very convincing description of an aggregate economy for it would imply that any additional income dYt earned in some period t is entirely spent in that period. In other words, there is no money being spent across periods. We now introduce a restriction on the MPC matrix M that rules out such strict “within period” spending.

Assumption 1.M is irreducible and noncyclic: (i) for each s, t 0, . . . , T there exists an m N such ∈ { } ∈ that (Mm) > 0; (ii) for each s 0, . . . , T ,M > 0. t,s ∈ { } ss Assumption1 has two parts. According to part (i), M needs to be such that, for any two periods s and t, an increase in aggregate income in period s will be partially spent in period t after m iterations. To give an example of m > 1, take a world in which additional aggregate income in period t + 1 is spent in period t but not period t 1. Yet, since spending in period t is equal to − income in period t, it is true—after two iterations—that some additional income in period t + 1

8While this assumption is always verified under a constant-r monetary policy rule, and we have verified it numerically in other cases, we currently do not have a proof that it is always verified.

22 will be spent in period t 1 despite the lack of a direct link. According to part (ii), additional − aggregate income earned in a given period will at least partially be spent in that same period. In describing these properties, we use terminology from the theory of Markov chains. In fact, when T is finite, the two conditions in Assumption1 establish the existence of a unique stationary distribution v of M, satisfying Mv = v, and the convergence of Mkw to the stationary distribution for an arbitrary intitial distribution w, as k ∞. → Assumption1 is verified in our benchmark model of section2, as well as, for example, in the two-agent New Keynesian model described by (30), except in the limit case where the fraction of hand to mouth agents becomes µ = 1. As we mentioned, an equation like the IKC (23) has a natural degree of indeterminacy, deeply rooted in the stochasticity of M: If dY is a solution, then so is dY + v whenever v is a vector that satisfies Mv = v. In cases with a finite horizon, T < ∞, such a vector v can only be equal to the stationary distribution of M (up to scale). When the horizon is infinite, however, this is no longer true as there can be cases in which v can no longer normalized to sum to 1. We therefore define the more general concept of a “regular” vector.

Definition 4. A vector v RT+1 is regular if and only if Mv = v. ∈ Intuitively, v can be thought of as “self-sustaining demand”: When the economy expects a path for incomes v , then agents find it optimal to demand v . At this point, it is worth reminding { t} { t} the reader that such self-sustaining demand v is inherent in many model with nominal rigidities and by no means special to our setup.9 In fact, as we shall see in the next sections, dealing with this indeterminacy is at the heart of many New Keynesian models. We are now in a position to precisely characterize the solution to the IKC (23). In particular, ∞ 10 the following result characterizes all possible solutions dY that have finite NPV ∑t=0 dYt.

Theorem 1 (Solving the Intertemporal Keynesian Cross). Let M satisfy Assumption1. Any finite- NPV solution dY RT+1 to the IKC (23) is given by ∈ ∞ dY = ∑ Mk∂Y + v, (31) k=0 where the infinite sum ∑∞ Mk∂Y is finite-valued and v RT+1 is a regular vector. Moreover, any k=0 ∈ dY RT+1 of the form in (31) is a solution to the IKC. ∈ Theorem1 justifies the “Keynesian Cross” part in the IKC. Whereas the solution to the Old- Keynesian cross, dY, is a scalar-valued infinite sum,

dY = ∂Y + MPC ∂Y + MPC2 ∂Y + ..., · · 9In fact, even if the economy has ﬂexible, but nominal prices, such a degree of indeterminacy still exists as all prices can be uniformly scaled up or down. 10There exist contrived examples when some solutions dY can have inﬁnite NPV, yet, as we discuss in the next section, those are not of interest to us.

23 the solution to the intertemporal Keynesian cross is a vector-valued sum

dY = ∂Y + M∂Y + M2∂Y + ... + v with an additional vector v that could potentially be nonzero. We already discussed how v captures indeterminacy due to self-sustaining demand, but what k does the ﬁrst term ∑k M ∂Y capture intuitively in the IKC? Consider a given PE shock ∂Y such as the government promising to purchase more in some periods and less in others. For simplicity, assume that M is transient (see the proof of Theorem1). To evaluate this term, consider its t-th element, which we can write as

T ∞ k ∑ ∑ (M )t,s ∂Ys . s=0 k=0 × |{z} | {z } PE income shock at time s total time t demand caused by time s income

This term is summing over all possible periods s. For each period s, it adds up the PE shock ∂Ys and the total spending effect it has on period t. Notice that this total effect can be very large (e.g. much larger than 1): Since one household’s spending is another household’s income, any PE shock

∂Ys may travel around the “MPC network” in various ways until it is spent in period t.

4.2 Determinacy

Solutions to the IKC are generally indeterminate, and so are linearized solutions to the general model of Section2. We now explain two ways in which there can be determinacy: First, in infinite horizon economies, it can be that there is only a single solution among all possible solutions that does not diverge in current values. Building on Woodford(2003) we call this solution locally determinate. Second, one can impose an ad-hoc rule that output dYt needs to obey, limiting the degree to which there can be multiplicity. As we show in AppendixD, these emerge naturally when the government’s budget constraint (10) is allowed to be violated off-equilibrium, so that there is fiscal dominance. For this section, we assume that the economy is at a steady state with a constant gross interest rate R > 1, and define r = log R > 0. In addition, we assume that the MPC matrix M does not asymptote to an identity matrix.

Assumption 2. If T = ∞, M is such that lim supt ∞ Mt,t < 1. →

Local determinacy in inﬁnite horizon economies. In an inﬁnite horizon economy, there can be situations in which the vector of self-sustaining demand v is unbounded in current value terms, rt that is, e vt is unbounded. Intuitively, any multiplicity would require agents to expect an ever increasing amount of income to generate enough demand for that income.

Deﬁnition 5. A vector w R∞ is bounded in current values if ertw is bounded in t. Otherwise, w ∈ t is unbounded in current values.

24 These deﬁnitions are useful to deﬁne the notion of local determinacy.

Deﬁnition 6. A solution to the IKC (31) is locally determinate (or locally unique) if it is the only solution dY that is bounded in current values. Otherwise, it is locally indeterminate.

Our definition of local determinacy follows the one in Woodford(2003), who introduced a similar notion of uniqueness in the context of the standard New-Keynesian model. There, it is shown that the introduction of a sufficiently responsive Taylor rule, that is, one that satisfies the Taylor principle, can generate local determinacy. Why would a Taylor rule help with determinacy, through the lens of our approach? According to our definition in (21), the Taylor rule is a part of the MPC matrix M: For example, when a sufficiently responsive Taylor rule is in place, any additional rise in income dYt earned in a period t is not merely spent according to the individual agents MPC’s across periods; it also raises real interest rates in period, incentivizing individuals to shift their spending into the future. Thus, a Taylor rule can direct the spending of any extra dollars earned in a given period into the future. Any such “spending in the future”, however, is precisely what requires self-sustaining demand to rt increase over time, possibly to the extent that e vt is unbounded. To characterize how much spending in the future there is after an increase in aggregate income, we define the following demand shift index λt∗.

Deﬁnition 7. Let M be an MPC matrix and deﬁne for each t 0 the characteristic function p : ≥ t R (0, ∞], → λ(s t) pt(λ) ∑ e− − Ms,t. (32) ≡ s

The demand shift index λt∗ of M at time t is deﬁned as the unique non-trivial solution to pt(λ) = 1, whenever such a solution exists, else we set λt∗ = 0.

λ(s t) To see that λt∗ is well-defined, observe first that pt(λ) is really the expectation of e− − under λ(s t) the probability distribution given by the t-th column of M. As e− − is convex in s for any t, pt(λ) is necessarily a convex function. Moreover, pt(0) = 1. Since M is irreducible, it must be the case that Ms,t puts positive weight on time periods other than s = t. Thus, the characteristic function pt(λ) is strictly convex (whenever finite) and therefore pt(λ) = 1 can only ever have a single solution other than λ = 0—thus, λt∗ is well defined.

Intuitively, λt∗ is a measure of how much of any additional income earned in period t would be spend in the future: When the t-th column (Ms,t)s puts a lot of weight on future time periods, λ(s t) this means coefﬁcients for e− − with s > t are large, pushing down pt(λ) for larger λ’s, which in turn raises λt∗. This is illustrated in Figure4: More weight on the future “tilts” pt(λ) to the right and therefore increases λt∗.

As it turns out, the demand shift index λt∗ is a powerful tool that allows us to summarize how much demand is pushed into the future of any additional aggregate income earned in a given period. We now relate λt∗ for far out time periods t to whether or not the equilibrium in our economy (and therefore the solution of the IKC) is locally determinate.

25 Characteristic function p∞(λ) 1.015

1.01

1.005

1 0.5 0 0.5 1 1.5 2 − − λ 10 2 · −

Figure 4: The characteristic function p∞ (λ) in our benchmark calibration with (Constant-r).

Theorem 2 (Local determinacy). Let M satisfy Assumptions1 and2. Suppose d Y is a solution to the IKC that is bounded in current values.

(a)d Y is locally determinate if lim inft ∞ λ∗ > r. → t −

(b)d Y is locally indeterminate if lim supt ∞ λt∗ < r. → −

A simple sufficient condition for (a) is lim inft ∞ pt( r˜) > 1 for some r˜ < r; a simple sufficient condition → − for (b) is lim supt ∞ pt( r) < 1. → − Theorem2 sharply characterizes when a given bounded solution dY is locally unique: Precisely when M tends to shift demand into the future to a sufficiently large degree. How large? Precisely r(s t) enough so that the sum of the current value spending pattern, p ( r) = ∑ e M , is strictly t − s − s t above 1 for all sufficiently large t. As we explain in detail in Section7, local determinacy in our general framework can also be induced by policies such as interest rate rules, if the economy is not already locally determinate. Foreshadowing these results, note that a more responsive Taylor rule for instance changes the M matrix, pushing the demand shift index λt∗ out to the right and shifting demand more into the future. According to Theorem2, larger demand shift indices always bring the economy closer to determinacy. To prove this remarkable result, we once more build on various results in Markov chain potential theory and on Foster-Lyapunov conditions to establish whether a nonzero self-sustatining demand vector v is bounded in current values or not. Intuitively, the proof works since the decay rate of any such v for large time periods is bounded below by lim inft ∞ λ∗ and above by → t lim supt ∞ λt∗. So when for instance lim inft ∞ λt∗ > r, then any such v must necessarily be → → − explosive in current values—and thus dY is locally determinate.

26 While this is a powerful local uniqueness result, one may wonder whether one can characterize the solution some more if it is indeed locally unique. As it turns out, for this it is necessary to distinguish MPC matrices M according to “how strongly” they shift demand into the future. Proposition 3 (Characterizing the locally unique solution). Let M satisfy Assumptions1 and2.

(a) If lim inft ∞ λt∗ > r and lim supt ∞ λt∗ < 0, the unique locally determinate solution dY (if it → − → exists) is given by ∞ k dY = 1 + va0 ∑ M ∂Y (33) k=0 where v is the unique (up to scale) stationary measure of M with eigenvalue 1, and is positive and bounded; and a RT+1 a unique vector (up to an additive constant). ∈

(b) If lim inft ∞ λ∗ > 0, the unique locally determinate solution dY (if it exists) is given by → t ∞ dY = ∑ Mk∂Y. k=0

∞ k In this case, the infinite MPC sum ∑k=0 M ∂Y can have nonzero net present value. Proposition3 establishes that, in the intertemporal Keynesian cross, the infinite MPC sum is generally not enough to characterize the locally unique output response dY. Indeed, the solution is a combination of the infinite sum and a self-sustaining demand term. There is, however an exception: If M pushes out demand in present value terms (case (b)), the locally determinate solution dY is just the simple infinite sum. Interestingly, the split in (a) and (b) according to how strongly M pushes demand out into the future is precisely reflected in whether the infinite sum can generate positive NPV responses by itself, or not. In appendix 29 we present a simple analytical example in which the economy falls into the “strong” case (b) so that the infinite sum is indeed sufficient to characterize the unique locally determinate solution.

Determinacy through ad-hoc rules. We have established local determinacy results for the uniqueness of solutions that are bounded in current values in infinite horizon economies. How- ever, in some cases, such as finite horizon economies or otherwise indeterminate economies, one may wonder whether other approaches to achieving determinacy exist that can be studied using our approach. We now explain one such approach, namely an ad-hoc approach, amounting to the imposition of an addition condition on dY. In AppendixD, we illustrate how such an ad- hoc constraint can be derived from the assumption of fiscal dominance in the fiscal theory of the price level. For simplicity, we explain the ad-hoc approach only for finite horizon economies. The analysis is similar in infinite horizon models.

Suppose the additional restriction on dY is given by a0dY = 0, where α is a T-dimensional vector, and T is the ﬁnite horizon of the economy. Then, the unique solution is described as follows.

27 Proposition 4 (Uniqueness with ad-hoc rule). The unique solution in a ﬁnite horizon economy with selection rule α0dY = 0 is given by

va T dY = 1 0 ∑ Mk∂Y. − a0v k=0

The easiest and arguably most intuitive selection rule is to set final period output to zero, dYT = 0. This mirrors local determinacy in infinite horizon economies, where at a locally determinate solution, dY 0 (since such a dY is bounded in current values). But alternatives exist. t → For instance, in some models of the fiscal theory of the price level, simple averages appear as well, 11 ∑t dYt = 0.

5 Determinacy and the new Taylor principle

5.1 The asymptotically self-similar IKC

As we have just seen, local determinacy of a GE response depends crucially on the far-out shape of the MPC matrix M. In this section, we consider an important subset of MPC matrices M, whose columns are asymptotically identical, just shifted down by one row. We call such M matrices self-similar, formally deﬁned as follows.

N N Definition 8. Let M be an infinite-dimensional matrix R × with well-defined characteristic functions p (λ) (see (32)). M is (asymptotically) self-similar if there exists a continuous func- { t }t tion p(λ), the asymptotic characteristic polynomial, such that p (λ) p(λ) as t ∞, pointwise for t → → all λ R . ∈ − Observe that we only require convergence for λ < 0 since more is not needed for the following result. In practice, however, most models generate MPC matrices where pt(λ) converges for all λ R. We can state the indeterminacy result for self-similar matrices. Define the scalar ∈

µ p ( r) ≡ − which corresponds to the area under the curve of the matrix of a far-out column of the M matrix, once expressed in current values relative to the time of the shock.

Corollary 2. Let M be a self-similar matrix satisfying Assumptions1 and2. In that case, the sequence of demand shift indices λ converges to a limit λ R. Moreover, let dY be a solution to the IKC that is t∗ ∗ ∈ bounded in current values. Then:

(a)d Y is locally determinate if λ > r or µ > 1. ∗ − (b)d Y is locally indeterminate if λ < r or µ < 1. ∗ − 11For papers using this approach to analyzing and resolving indeterminacy, see Leeper(1991), Woodford(1995), Cochrane(2011), or Caramp and Silva (2017).

28 Far-out column of M matrix Far-out column of M matrix 0.15 0.15 PV PV CV

0.1 0.1

Area = 1 Area = µ 0.05 0.05

0 0 30 20 10 0 10 20 30 30 20 10 0 10 20 30 − − − − − − Year relative to shock Year relative to shock

Figure 5: Simple criterion for multiplicity: µ vs 1

Self-similar MPC matrices play a prominent role because models with idiosyncratic risk and incomplete markets, such as our main model of section2, naturally generate such MPC matrices. Figure5 illustrates this criterion in the context of our benchmark model under a (Constant- r) rule. The left panel shows a column of M = MY for a shock at some date T 0, centered around date T. By construction, since ∑ Ms,T = 1, we know that the area under this curve is 1. The right s T panel plots these in current values relative to the time of the shock The area µ = ∑ Ms,T (1 + r) − may in principle be greater or lower than 1 depending on the balance of weights to either side of T. When µ > 1, weights tend to fall towards the future (“demand is pushed to the future”), generating determinacy. The criterion is very intuitive and broadly applicable to models with incomplete markets as a simple test of determinacy. It also yields intuitive comparative statics, as described below.

5.2 A New Taylor Principle

So far, we have provided generic conditions for determinacy that do not exploit speciﬁc features of the M matrix. We apply these below to understand determinacy in an economy under (Constant- r), for which lemma5 showed that M = MY. We now consider the consequences of the structure imposed by monetary policy that follows (6), as described in6. R λ(s t) R Let p (λ) limt ∞ ∑s e− − M , bet the asymptotic characteristic polynomial of the (real) ≡ → s t interest rate response matrix MR. By construction, pR(0) = 0. Moreover, typically, higher interest rates push demand into the future, so that pR( r) > 0. −

29 We obtain the following generalized Taylor principle. Deﬁne

µY = pY( r) − µR = pR( r) −

Theorem 3 (Generalized Taylor Principle.). Let M be a self-similar matrix satisfying Assumptions1 and2. Let MR be a self-similar matrix whose columns sum to zero. Assume Mjoint is similar to a (non- negative) column-stochastic matrix. Then, 1 β 1 µY (a) if φ , φ are such that φ > 1 − φ + − any equilibrium is locally determinate; y π π − κ y µR 1 β 1 µY (b) if φ , φ are such that φ < 1 − φ + − any equilibrium is locally indeterminate. y π π − κ y µR In particular, when the Taylor rule is only based on inﬂation, φy = 0, the threshold responsiveness φπ for 1 β 1 µY determinacy is equal to 1 − φ + − . − κ y µR Our result in Theorem3 succinctly summarizes when how strong a Taylor rule is required to be in order to induce determinacy in any macroeconomic model that can be described by a self- similar MPC matrix M and a self-similar interest rate response matrix MR. It stresses the role of Y R the values µ and µ . We explain the intuition behind both for the simple case where φy = 0. The position of µY relative to 1 is a measure of how much demand is shifted into the future. In the last subsection, we saw that under a constant real interest rate, 1 is precisely the threshold µY has to exceed for determinacy. Theorem3 adds to this result that where µY lies relative to

1 precisely determines whether the Taylor rule has to respond to inflation φπ more or less than one-for-one. If µY is large compared to 1 and prices are not too flexible, that is, κ is small, it may even be that the economy is determinate with a Taylor rule that does not respond to inflation at all, φπ = 0. The magnitude of µR roughly captures how responsive the economy is to real interest rates. To illustrate why this is important, imagine an economy with µY < 1. How responsive does the Taylor rule have to be to shift enough demand out into the future and generate determinacy? This is exactly where µR matters: if the economy’s responsiveness to real interest rates µR were large

(say inﬁnite), even a small increase in φπ relative to 1 will be enough to ensure determinacy. Thus, R Y Y the threshold for φπ declines in µ when µ < 1, and vice versa if µ > 1.

5.3 Determinacy with constant-r in our model

We start by considering the case of our model with monetary policy described by (Constant-r). In this case, the position of µY relative to 1 matters. Figure6 shows the value of µY in our model under various assumptions about two key parameters: the degree of “cyclicality of income risk” ∂2 log Γ b γ = ∂ log l∂ log e and the amount of aggregate liquidity y . The dot on each curve represents our benchmark calibration detailed in table1, with Γ (e, l) = l b Y so that γ = 0 and y = 140%. In this benchmark, µ = 1.08, so that the economy with constant- r is determinate. This therefore contrasts with the case of a representative agent, for which the

30 1.2 1.2 µY 1.15 Main calib. 1.15

1.1 1.1

1.05 1.05

Determinacy 1 1 Indeterminacy 0.95 0.95 0.6 0.4 0.2 0 0.5 1 1.5 2 2.5 3 − − − Cyclicality of income risk γ Liquidity to GDP ratio b/y

Figure 6: Determinants of determinacy: Cyclicality of income risk and liquidity. economy is just indeterminate with constant real interest rates. The intuition is that, relative to that case, agents in our benchmark incomplete markets model tend to spend incomes after they receive it, in large part because of borrowing constraints. The intuition that more stringent borrowing constraints always push towards determinacy is incorrect, however. Consider the left panel, which illustrates what happens as we make income risk more countercyclical, taking γ to more negative values. In this case, agents with low skills e are disproportionately hurt when employment l falls. Suppose that agents expect employment to fall in future periods. Then they get collectively worried about the future and end up cutting consumption today as well. Symetrically, if they expect the economy to improve, they cut precautionary savings and increase their spending immediately. The outcome is an economy that tends to spend in anticipation of income—far-out M columns are tilted towards the past, and this tends to generate indeterminacy with (Constant-r). Figure7 further illustrates this in two extreme cases: one indeterminate case with γ = 0.75 and one determinate case with γ = 0. Observe that the − impulse responses in the former case are more tilted to the past than those in the latter case. The resulting multiplicity vector v (plotted in current value terms of the right panel of ﬁgure7), is explosive under γ = 0 but converges to zero under γ = 0.75. What self-sustains this vector in − that case is the following feedback mechanism: an expectation of improvement of the economy tomorrow generates even more spending today, so the pattern of demand over time self sustains very precisely. This feedback loop between precautionary savings and the level of economic activity under nominal rigidities has been highlighted by several authors before (see Ravn and Sterk 2013, den Hann et al. 2015, Bayer et al. 2015, Challe et al. 2014, or Heathcote and Perri 2016), but had never been related to a matrix of MPCs until now. The right panel of ﬁgure6 plots another interesting comparative statics: that of the level of b b liquidity y . For each value of y on the left axis, we recalibrate the value of β to hit the constant interest rate r = 4%, and then plot the resulting µY. Here, it is always the case that µY > 1 so that

31 Impulse response to far-out income shock Multiplicity vector 0.2 γ = 0.75 γ =− 0 2 0.15

0.1 1.5

0.05 1

0 0.5 5 0 5 10 0 5 10 15 20 − Time relative to shock Time

Figure 7: Understanding multiplicity. the economy is determinate. Further, µY smoothly converges to 1 as liquidity disappears and all agents become nearly completely constrained, with MPCs equal to 1 for each of them.

5.4 Applying the New Taylor Principle

We next turn to the case when monetary policy in our model is described by (Taylor rule). We consider a case where φy = 0 and plot

1 β 1 µY φ∗ = 1 − φ + − π − κ y µR for the two comparative statics of the model described in the previous section, maintaining the calibration of table1 except as explicitly mentioned. The left panel of figure6 illustrates the same force at play: across all these experiments µR is constant, and therefore, the more countercylical income risk is (the more negative γ), the higher φ needs to be to ensure determinacy. When γ = 0.75 (the extreme case to the left of the graph), y − the Taylor rule coefficient needs to be almost 1.1 to ensure determinacy. This illustrates the fact that monetary policy has to fight off the natural tendency of the economy to indeterminacy, by responding more strongly in terms of interest rates. By constrast, the right panel of figure8 illustrates a new force: as the economy has less and less liquidity, agents become almost completely unresponsive to changes in real interet rates, and further µR limits to zero faster than µY limits to 1. Hence the economy behaves in a determinate way, as if r was constant, even if monetary policy varies the nominal interest rate substantially in response to developments in inflation. A conclusion from this analysis is that important factors are likely to affect the degree to which

32 1.1 φπ∗ Main calib. 1 1.05

1 0 0.95

0.9 1 − 0.85 2 0.6 0.4 0.2 0 − 0.5 1 1.5 2 2.5 3 Taylor rule determinacy threshold − − − Cyclicality of income risk γ Liquidity to GDP ratio b/y

Figure 8: Determinants of the Taylor rule thresholds. monetary policy needs to lean against the wind to ensure economic stability. For example it has been widely argued (Clarida, Galí and Gertler(2000)) that monetary policy has small responses to inﬂation in the pre-1980 period. However, to the extent that the amount of liquidity at the time was also limited, ﬁgure8 suggests that this may not have been a cause of instability. A further exploration of this topic is an interesting question for future research.

6 When are shocks a net stimulus?

Consider the response to a government spending shock in our benchmark model. The government spends an extra unit today, and simultaneously increases debt, withdrawing it at a rate of 20% per period.12 Figure9 shows the outcome of this experiment. In partial equilibrium, households cut consumption, with some anticipation of the taxes and a large effect at the time the taxes are levied–an effect that fades out over time. As per lemma3, the net present value of the impulse reponse is clearly 0. However, in general equilibrium, the impact effect is greater that 1, and further persists for a number of periods. It never goes below 0. This experiment lays bare one of the most important questions one may ask about aggregate demand: Where does it come from? How can it be, that a zero net present value partial equilibrium shock ∂Y causes a nonzero net present value general equilibrium response dY? What features of the MPC network M make this possible? And what features of ∂Y decide whether dY has positive net present value vs. negative net present value?

12 t Speciﬁcally, dg0 = 1, db0 = 1, dbt+1 = ρ dbt for ρ = 0.2 and t > 0.

33 Partial Equilibrium General Equilibrium 1.5 1.5 Output demand Output

1 1

0.5 0.5

0 0

0 2 4 6 8 10 0 2 4 6 8 10 Time Time

Figure 9: Fiscal policy: the effect of government spending with delayed taxes

6.1 Theory

The crucial “symmetry breaking” ingredient to answering these questions is going to be time: We show that if the MPC matrix is ordered, in that later increases in aggregate incomes are spent later that earlier ones, then a partial equilibrium impulse ∂Y causes a positive NPV GE response dY precisely if it is front-loaded; and vice-versa a negative NPV response dY if it is back-loaded. We deﬁne these concepts formally as follows.

Deﬁnition 9. Let M R(T+1) (T+1) be a column-stochastic matrix, and let v , v RT+1 be two ∈ × 0 ∈ vectors.

> a) M is ordered, if for any t0 t, (Ms,t0 )s ﬁrst-order stochastically dominates (Ms,t)s.

b) v0 is more front-loaded than v, if for any t, ∑τ t vτ0 ∑τ t vτ. In this case, v is more back- ≤ ≥ ≤ loaded than v0.

c) v0 is front-loaded if it is more front-loaded than a vector of zeros. v is back-loaded if it is more back-loaded than a vector of zeros.

This allows us to state the following main result on the origins of aggregate demand.

Proposition 5 (Ordered matrices and font-loading). Let M be an ordered matrix satisfying Assump- tions1 and2. Let d Y and dY0 be locally unique GE responses to PE impulses ∂Y and ∂Y0. If ∂Y0 is more front-loaded than ∂Y, then dY0 is more front-loaded than dY.

In particular, if ∂Y is front-loaded, dY has non-negative initial impact dY 0 and non-negative net 0 0 ≥ present value ∑ dY 0. The opposite holds if ∂Y is back-loaded: dY 0 and ∑ dY 0. t t ≥ 0 0 ≤ t t ≤

34 The proposition makes clear what the key feature is that generates nonzero net present value aggregate demand: how front-loaded the partial equilibrium shock is ∂Y. In particular, the only difference between a contractionary and an expansionary PE shock is how front-loaded they are, not their net present value (which is zero for both). Of course, Proposition5 requires the MPC matrix M to be ordered. While we have not been able to obtain an analytical proof for the general model of section2, it is a condition that we have easily checked to be true in all our analytical and numerical applications, and that we conjecture to be true more generally.

6.2 IKC iterations and tatonnement to equilibrium

Figure 10 illustrates a ’tatonnement’ process that happens to deliver general equilibrium as an out- K k come in the fiscal multiplier scenario of figure9: it plots the partial sums ∑k=0 M ∂Y for various K’s. In this case, this process converges to the general equilibrium outcome dY. This corresponds to case b) in theorem3, for which the Markov chain represented by M is transient and iterations converge to the general equilibrium outcome. In this process, we see that the negative demand values initially generated by the partial equilibrium effect (owing to the taxes being levied) ultimately never materialize. The intuition is as follows: as agents receive more income in period 0 due to higher government spending, they spend this additional income in period 0 and 1. This mitigates the fall in demand due to the lower taxes in period 1, ultimately raising income in period 1, which is then spent in period 2, and so on. Once the intertemporal keynesian cross has run its course, the initial negative demand has been pushed out to infinity and is not featured at all in the general equilibrium impulse response.13

7 When does heterogeneity matter for ﬁscal and monetary policy?

In this section, we conclude by using our new tools to answer an important question in the literature: when does heterogeneity matter for the aggregate effect of monetary and ﬁscal policy? We do so by exhibiting two benchmarks for which heterogeneity does not matter, in the sense that a ﬁscal shock or a monetary shock delivers the same answer as under the representative agent case. By looking how, in partial equilibrium, a given economy departs from a benchmark, we are able to characterize the precise conditions under which heterogeneity matters.

7.1 Fiscal policy

We start by deﬁning a balanced budget ﬁscal policy.

13Note however that, in case a) of theorem3, which features a recurrent Markov chain—so that the economy’s natural tendency to push demand into the future is not as strong—this speciﬁc process will not converge to the unique equilibrium.

35 From partial to general equilibrium 1 K = 2 0.8 K = 5 K = 30 0.6 GE (K = ∞)

0.4

0.2

0.2 − 0 10 20 30 40 50 Time

Figure 10: From PE to GE: Demonstrating the iteration.

Deﬁnition 10. A balanced budget government spending shock is a shock to government spending dgt that does not lead to a change in debt at any point in time: dbt = 0 for all t. In equilibrium, the government adjusts taxes and transfer to raise revenue in the period of the shock, drevt = dgt.

Proposition 6 (Benchmark multiplier of 1). Assume a) equal incidence of both gross Γ (e, l) = l, b) equal incidence of net incomes ϕ = τr, and c) monetary policy follows (Constant-r). Then the general equilibrium effect of a balanced-budget increase in government spending is equal to the path of spending itself. This is also the representative agent response.

dY = ∂G

Proof. Let gt be a new path for spending. Conjecture an equilibrium in which the entire stochastic process for agent post tax incomes zt (eit) remains unchanged at their initial level, for every t. Since monetary policy follows (Constant-r), the real interest rate is also constant for all agents. Preferences and borrowing constraints have not changed, and therefore each agent has an identical consumption path, so aggregate consumption path follows the same path as before. The government’s additional revenue requirement is drevt = dgt at every t. Under our conjecture, r GDP also changes by dyt = dgt at every t. Hence, the fiscal rule (12) with ϕ = τ implies that r the lump sum tt is unchanged at every t. Moreover, with ϕ = τ the fiscal rule (11) also implies that tax revenue d (τtyt) = dyt = dgt. Total tax revenue goes up exactly by enough to pay for the extra government spending. Putting this together and using equal incidence Γ (e, l) = l, agent incomes all change by dz = 0 + e d ((1 τ ) y ) = e (dy d (τ y )) = 0, confirming the initial it it − t t it t − t t equilibrium.

36 Partial Equilibrium General Equilibrium 1.5 1.5 Balanced-budget G

1 1

0.5 0.5

0 0

0.5 0.5 − 0 5 10 15 20 − 0 5 10 15 20 Time Time

Figure 11: Fiscal policy: balanced-budget multiplier

Proposition6 delivers a benchmark of one for the fiscal multiplier—a surprising result, given all of the heterogeneity. This is true no matter what the path for government spending is. As is well known, this benchmark of 1 is the same as the one that obtains in the standard New Keyne- sian model under a constant-r monetary policy (see, for example, Woodford(2011)), and would of course obtain under the representative agent version of our model. Hence, in this case, heterogeneity is irrelevant for understanding the general equilibrium effect of fiscal policy. A recent literature has computed fiscal multipliers in heterogeneous agent models (see for example Hagedorn et al.(2017)), finding multipliers sometimes smaller and sometimes larger than 1. Proposition6 establishes exactly when a multiplier of 1 obtains. The next section then quantities a force that can lead to larger multipliers.

Balanced budget vs. delayed taxes. Consider a government spending shock financed by delayed taxes, such as that of figure 11. Using the linearity of our operator, it is natural to decom- G pose this shock as the combined effect of a balanced budget fiscal shock (which has multiplier of 1) ∂Gbb and an effect from a tax rebate today, with higher taxes in the future ∂Ctaxrebate. Applying linearity and proposition 1, we know that dY = ∂G + ∂Ctaxrebate. Applying proposition5, we G further know that—since a tax rebate shock has frontloaded spending—the effect of ∂Ctaxrebate is G positive. The more the taxes are delayed, the greater the fiscal multiplier, as illustrated in figure 12. Quantitatively, this shows that departures from Ricardian equivalence can be largely amplified in general equilibrium.

7.2 A benchmark result for monetary policy

Next, we define a new benchmark for monetary policy shocks. This benchmark corresponds to what we label the modified substitution effect. [...to be defined. Proposition: the modified sub-

37 Partial Equilibrium General Equilibrium 1.5 1.5 ρ = 0.2 ρ = 0.8 1 ρ = 0.95 1

0.5 0.5

0 0

0.5 0.5 − 0 2 4 6 8 10 − 0 2 4 6 8 10 Time Time

Figure 12: General equilibrium effect of increasingly delaying taxes stitution effect is the representative agent effect. This relates to Werning(2015), but is different because it does not require his case of σ = 1 or zero liquidity, under which the modified income effects cancel.]. We can also see how the modified substitution effect corresponds to the more standard substitution effect of interest rate changes. In partial equilibrium, using Slutsky’s equations, we define the subtitution effect as ∂Y = ∂Cinc + ∂Csub we then consider the general equilibrium effect, in turn for contemporaneous monetary policy shocks and for forward guidance shocks.

Standard monetary policy shocks. Standard monetary policy shocks operate, in our model, both due to income and to substitution effects. In general equilibrium, the substitution effects are almost exactly the representative agent response, and the income effects contribute to amplify the response by around 50%. This conﬁrms the results in Auclert(2017), who argued that income effects were likely to operate in the same direction as subtitution effects and to amplify monetary policy under heterogeneous agents.

Forward guidance shocks. Figure 14 repeats the exercise for a forward guidance shock: a change in interest rates at date t = 10. Here, we ﬁnd again that the income effect is what makes the difference to the representative agent case, but this time in the other direction, generating dampening of forward guidance in general equilibrium. The reason here is as follows: consider a contractionary monetary policy shock that will raise interest rates in the future. Agents who will beneﬁt from this increase in interest rates tend to be relatively unconstrained savers, who can immediately increase their consumption in response. By contrast, agents who will lose from this

38 Partial Equilibrium General Equilibrium 0.1 0 0

0.2 0.1 − −

0.2 − 0.4 income − 0.3 substitution − overall 0.6 0 2 4 6 8 10 − 0 2 4 6 8 10 Time Time

Figure 13: Decomposing standard monetary policy shocks increase in interest rates tend to be more constrained, and do not lower their consumption further. This generates an asymmetry for the income effect that results in an overall dampening of monetary policy.

8 Conclusion

[To be added. ]

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A Union wage setting

A.1 Details of the labor market

We assume that the worker provides nijt hours of work to each of a continuum of unions indexed by j [0, 1], so that his total labor effort is ∈ Z 1 i nit njtdj ≡ 0

The agent gets paid a nominal market wage of wjt per efﬁcient unit of work in union j. Hence his total nominal gross earnings are Z 1 wteitnit wjteitnijtdj ≡ 0

Each union j aggregates efﬁcient units of work into a union-speciﬁc task ljt, such that

ljt = ∑ π (eit) eitnijt (34) ei

Aggregate employment lt, in turn, is an aggregate of union-speciﬁc tasks that are imperfect sub- stitutes for one another, e Z 1 e 1 e 1 −e − lt ljt dj ≡ 0

42 Each union sets its own wage wjt per unit of task provided. We assume that nominal wages wjt are partially rigid, and only reset occasionally in a Calvo fashion: each period, any given union j keeps its wage constant with probability θ. At every point in time, the wage index

1 Z 1 1 e 1 e − wt = wjt− dj 0 serves as a reference for ﬁnal goods ﬁrms’s demand for tasks. Conditional on employment lt, this demand is e wjt − ljt = lt wt

In turn, union j, when producing ljt tasks, pays all of its workers the same wage wjt and demands that those with skill eit work nijt hours according to a particular rule Γ, so that

nijt = Γ eit, ljt (35)

The Γ function must be consistent with aggregation (34) for any ljt. A simple example is Γ (e, l) = l.

In this case, each union member works nijt = ljt hours irrespective of their skill or their consumption level. A wage-resetting union maximizes an average of the utilities (3) of each of its workers, with each worker receiving weight λi. In appendixA, we write down the wage setting problem of the union conditional on aggregate employment l and a real wage w . We show that in a steady state t pt in which employment is constant at l∗, prices are constant at p∗ and all resetting unions set wages at w∗, the consumption distribution Ψ (c) statisﬁes

i dv (Γ(e,l )) E λ (e, c) Γ (e, l ) l dn ∗ w e e,Ψ(c) l ∗ ∗ e ∗ = (36) " Γ(e,l ) # p∗ e 1 e ∗ − E R − Γl (e,l∗)l∗ du λ (e, c) γl (e, l∗) l∗ e 1 dc (c) −

Intuitively, union wage-setting ensures that the real wage w∗ is at a markup over some average p∗ across union members of the marginal rate of susbtitution between consumption and hours. The wage setting rule (36) requires knowledge of the consumption distribution in the popu- i lation c∗ . Away from steady-state, we assume that unions reset wages under a behavioral approximation that this distribution remains constant. The underlying assumption is that it is too complicated for the union to keep track of each member’s asset position over time when that distribution is changing. Practically, this implies that the union ignores income effects on labor supply when resetting wages away from steady-state. Below we show that this problem leads to a simple New Keynesian Phillips curve for aggregate wage growth, πw log Wt , given to ﬁrst t Wt 1 ≡ − order by 1 πw = λ lb (w p ) + β πw (37) t t ψ t − t − t t t+1

43 where l log (l /l ), ψ is a constant, and κ , β are deterministic functions of time, with β = β t ≡ t ∗ t t t when = ∞. T

A.2 Derivation of the general case

Consider the problem of a union j. The union supplies ljt units of efﬁcient work by aggregating labor from each of its members according to (34), by allocating work across its living members i i i acording to the rule in (35), that is etnt = γ ljt, et . Consider ﬁrst the general case. Given a path for agregates lt, Pt, Wt , wages at every union { }i Wjt , and an assumed distribution for the consumption of every agent ct , the problem that the union k solves when it gets a change to reset its wage Wkt is

Z Z τ i i i i i max ∑ (βθ) λ u ct+τ; θ v njt+τdj; θ Wkt τ 0 − ≥ where λi is the weight assigned to member i in social utility. The ﬁrst-order condition for this problem is

Z dui i τ i dc ct+τ i i ∑ (βθ) λ γ lkt+τ, et+τ eγl lkt+τ, et+τ lkt+τ di τ 0 Pt+τ − ≥ dvi R i Z dn njt+τdj + τ i i = ∑ (βθ) λ e i γl lkt+τ, et+τ lkt+τdi 0 τ 0 et+τWkt ≥ which we can rewrite as

Z i τ i du i i ∑ (βθ) λ ct+τ γl lkt+τ, et+τ lkt+τ τ 0 dc ≥  i  i ! dv R i γ l + , e W dn njt+τdj e kt τ t+τ kt e = 0  i i  di − γ l , e l Pt+τ − du i i l kt+τ t+τ kt+τ dc ct+τ et+τ

Hence all resetting unions choose the same wage Wkt = Wt, satisfying the implicit equation

i R dv ni dj τ R dn jt+τ (βθ) λiγ l , ei l di e ∑τ 0 l kt+τ t+τ kt+τ ei W = ≥ t+τ (38) t  i  e 1 γ(lkt+τ,et+τ ) e i − − γ l ,ei l du ci τ R i i l ( kt+τ t+τ ) kt+τ dc ( t+τ ) ∑ (βθ) λ γ l +τ, e l +τ   di τ 0 l kt t+τ kt  e 1  Pt+k ≥ −

44 In the special case where γ is linear in l, so that γll = γ,(38) simpliﬁes to

i R dv ni dj τ R i i dn jt+τ ∑τ 0 (βθ) λ γ lkt+τ, et+τ i di e ≥ et+τ Wt = i du ci e 1 τ R i i dc ( t+τ ) − ∑τ 0 (βθ) λ γ lkt+τ, et+τ P di ≥ t+k

In a steady-state in which the price level is P∗, the nominal reset wage is equal to the constant i W = W∗, labor is the constant l∗, and the consumption distribution is c∗ ,(38) yields

dvi 1 γ , i i Z dn ei l∗ e W Z du i i = ∗ i i i i e λ γl l∗, e l∗ i di λ eγl l∗, e l∗ γ l∗, e c∗ di (39) e P∗ − dc which is equation (36). In general equilibrium, equation (39) implicitly deﬁnes the level of steady- state employment l∗, where the consumption distribution is consistent with a real earnings process W∗ i given by γ l∗, e . We linearize around such a steady state. P∗ From (38), we can write the reset wage relative to the current wage Wt as

i R dv ni dj τ R i i dn jt+τ e ∑τ 0 (βθ) λ γl lkt+τ, et+τ lkt+τ i di Wt ≥ et+τWt Gt = i = (40) du ci Wt τ R i i i dc ( t+τ ) Ht ∑τ 0 (βθ) λ eγl lkt+τ, et+τ lkt+τ γ lkt+τ, et+τ P di ≥ − t+k where Gt, Ht satisfy the recursions dvi R i Z dn njtdj W = i i + t+1 Gt e λ γl lkt, et lkt i di βθ Gt+1 (41) etWt Wt Z dui i i i i dc ct Ht = λ eγl lkt, et lkt γ lkt, et di + βθHt+1 (42) − Pt

The wage index Wt at time t satisﬁes the recursion

1 e 1 e 1 e Wt − = θWt −1 + (1 θ)(Wt∗) − (43) − − Take logs on both sides of (43) and (40) and rearrange to get

w 1 θ 1 θ πt wt wt 1 = − (wt∗ wt) = − (gt ht) (44) ≡ − − θ − θ −

Consider the steady state where lt = l∗, Pt = P∗, Wt = Wjt = W∗ for all j and the distribution i i of consumption is at its steady state c∗ . From (35), all workers with productivity et work equally at each ﬁrm, γ l , ei i = ∗ t njt i j et ∀

45 Then (41)–(42) deliver

i dvi γ(l∗,e ) Z dn ei = i i Gt∗ Φte λ γl l∗, e l∗ i etW∗ Z dui i i i i dc c∗ Ht∗ = Φt λ eγl l∗, e l∗ γ l∗, e di − P∗

T t τ where Φt = ∑τ=−0 (βθ) . Using (39), we obtain

Gt∗ = CΦt

Ht∗ = CΦt

dvi 1 i i γ l∗,e du i R i i dn ei ( ) R i i i dc (c∗ ) with C = e λ γl l∗, e l∗ i di = λ eγl l∗, e l∗ γ l∗, e di. e W∗ − P∗ Gt Ht Approximating (41)–(42) around Gt∗, Ht∗, and deﬁning gbt log G and gbt log H , we ≡ t∗ ≡ t∗ obtain

1 1 gbt = at + 1 (wt+1 wt + gdt+1) (45) Φt − Φt − 1 1 hbt = bt + 1 hdt+1 (46) Φt − Φt Where we have deﬁned ( ! ) Z γ l , ei l 1 γ l , ei l Gi ll ∗ ∗ + + + l ∗ ∗ at ω i 1 e (wkt wt) lbt i i i lbt wt di ≡ γl (l∗, e ) − − ψ (e ) γ (l∗, e ) −

In this expression, ψi is the Frisch elasticity of labor supply of individual i in state ei, and individual weights are the following transformation of social weights:

! γ l ,ei dvi ( ∗ ) dn ei i i Gi λ γl l∗, e l∗ ei ω = i dv 1 γ(l ,ei) R i i dn ei ∗ λ γl (l∗, e ) l∗ ei di

Similarly,

Z ( i ! i !) ! e γll l∗, e l∗ 1 γl l∗, e l∗ 1 b = ωHi + 1 e (w w ) + lb cbi p di t e 1 γ (l , ei) − e 1 γ (l , ei) − kt − t t − σ (ei) t − t − l ∗ − ∗ with i i i dui i λ eγ l∗, e l∗ γ l∗, e c∗ ωHi = l − dc R i λi (eγ (l , ei) l γ (l , ei)) du (c i) di l ∗ ∗ − ∗ dc ∗

46 R Hi 1 bi = Our behavioral assumption for the union consists in assuming that ω σi(ei) ctdi 0, that is, the consumption distribution remains at its steady-state. Using this assumption to simplify bt, we can solve for the difference ! Z 1 γ l , ei l = Gi l ∗ ∗ at bt ω i i i di lbt (wt pt) − ψ (e ) γ (l∗, e ) − − Z ( i !) ! 1 γll l∗, e l∗ + ωGi − di lb e (w w ) e 1 γ (l , ei) t − t − t − l ∗ 1 a b = l (w p ) eν (w w ) (47) t − t ψ t − t − t − t − t where 1 Z 1 γ l , ei l Gi l ∗ ∗ + ω i i i di ν ψ ≡ ψ (e ) γ (l∗, e ) i ! Z 1 γ l∗, e l∗ ν ωGi − ll di ≡ e 1 γ (l , ei) − l ∗ We ﬁnally obtain the wage Phillips Curve as follows. Start with equation (44)

w 1 θ 1 θ 1 θ πt = wt wt 1 = − (wt wt) = − (gt ht) = − gt hbt (48) − − θ − θ − θ b − where the last line follows Gt∗ = Ht∗. Next, combine (48), (45), (46) and (47) into

w w = g hb t − t bt − t 1 1 = (at bt) + 1 wt+1 wt + gdt+1 hdt+1 Φt − − Φt − − 1 1 1 = lt (wt pt) eν (wt wt) + 1 wt+1 wt + gdt+1 hdt+1 Φt ψ − − − − − Φt − − solve out for w w t − t 1 eν − 1 1 1 wt wt = 1 + lt (wt pt) + 1 wt+1 wt + gdt+1 hdt+1 − Φt Φt ψ − − − Φt − −

w Wage inﬂation πt is then

1 1 θ eν − 1 1 1 1 1 θ wt wt 1 = − 1 + lt (wt pt) + 1 1 (wt+1 wt) + − gdt+1 hdt+1 − − θ Φt Φt ψ − − − Φt θ − − θ − 1 eν − 1 1 θ 1 1 1 = 1 + − lt (wt pt) + 1 (wt+1 wt) Φt Φt θ ψ − − − Φt θ −

This gives us our ﬁnal expression for the wage Phillips curve

47 1 πw = λ lb (w p ) + β πw t t ψ t − t − t t t+1 where

1 eν − 1 1 θ λt 1 + − ≡ Φt Φt θ 1 1 βt 1 ≡ − Φt θ T t − τ Φt ∑ (βθ) ≡ τ=0

1 which is equation (9). Note in particular that Φt 1 βθ as T ∞, and therefore βt β. → − → →

B Alternative economies

B.1 OLG economy

Consider an inﬁnite-horizon model with overlapping generations of agents living for two periods each. At every time t live two generations of equal size: an old generation born at time t 1, and − a young generation born at time t. The old consume their savings st. The young work nt hours at a nominal market wage Wt, pay real taxes tt, and can buy consumption goods with price Pt or save in nominal bonds with price 1 . Each young person born at time t solves the maximization 1+it problem

max u (ct,t) + βu (ct,t+1) st+1 Ptct,t + = Wtnt Pttt 1 + it − Pt+1ct,t+1 = st+1

1 1 where u has constant elasticity of substitution σ, u (c) = c − σ . Only the current young supply 1 1 − σ labor, and they work as many hours n as ﬁrms demand given the real wage Wt . For the purpose t Pt of this illustrative example, we assume that wages are perfectly rigid at a constant level (θ = 1), and normalize this level to 1 for convenience:

Wt = 1

Hence ﬁrm price setting (8) implies that the nominal price level is constant at Pt = 1. This equation also implies that the nominal and the real interest rate are equal at all times, Rt = 1 + it. We assume that monetary policy sets a constant path for the nominal interest rate i equal to β 1 1. This is t − − equivalent to our (Constant-r) assumption.

In each period, the government only levies taxes tt on the current young . Its budget constraint

48 is therefore bt+1 + Pttt = bt + Ptgt (49) Rt

At t = 0 one generation is born old, and the government endows these agents with b0 bonds. Given a path for government policy b , g , t , i satisfying (49) at each point in time and the { t t t t} terminal condition ∞ 1 b = 0, as well as a ﬁxed level for the nominal wage W = 1, an ∏s=0 Rs ∞ t equilibrium in this economy is a path for prices Pt, Rt , and quantities c 1,0 and yt, n, ct,t, ct,t+1 , { } − { } such that households optimize, ﬁrms optimize, and the goods and bond markets clear:

gt + ct,t + ct 1,t = yt − st = bt

Solving for equilibrium. Given a feasible path for policy b , g , t , i , equilibrium is deter- { t t t t} mined as follows. Household’s optimal intertemporal choice given the real wage Wt = 1 and Pt 1 the real interest rate Rt = β− determines the path of aggregate savings st+1,

st+1 1 = nt tt (50) Pt+1 1 + β { − }

st+1 bt+1 Bond market clearing imposes that P = P , while the government budget constraint requires t+1 t+1 bt+1 1 bt = + gt tt . Solving out these equations for yt, we obtain the equilibrium output and Pt+1 β Pt − employment level as a function of current government policy and the outstanding debt stock, 1 bt yt = nt = (1 + β) + gt tt t (51) β Pt − ∀

Equation (51) shows neatly how output is determined by government policy. Examining equation (51) leads us to a number of important remarks.

Equilibrium determinacy. Notice that equilibrium is uniquely pinned down in our economy, despite the fact that the path of nominal interest rates is constant. This negates a classic result in monetary economics, dating back to Wicksell(1898) and more recently Sargent and Wallace(1975), according to which equilibrium is indeterminante under a pegged interest rate. Section 4.2 will shed light on this rather mysterious result.

Government spending multipliers: benchmark is 1, higher if frontloaded. Equation (51) shows that an increase in current spending gt paid for by current taxes tt increases output one-for- one: the tax-ﬁnanced spending multiplier is one.

dy t = 1 (52) dg t dgt=dtt

49 Moreover, an increase in current spending paid for by future taxes has more than one-for-one effect on current output. It also has an effect on future output, since the current government deﬁcit increases future debt. Indeed, along a path with dtt = 0, it is immediate to show that

dy 1 k+1 t+k = (1 + β) > 1 (53) dg β t dtt=...=dtt+k=0

Of course, this process cannot go on indeﬁnitely since the government must eventually raise taxes to satisfy its intertemporal budget constraint. However, this observation illustrates a key property of equilibrium multipliers, which is that they are raised by frontloading spending relative to taxes. In section 6.1, we provide a substantial generalization of both of these results.

The intertemporal Keynesian Cross. It is simple to show that the present value impulse re- t sponses dYt = β dyt must satisfy the following equilibrium conditions

1 dY = dY dT + dG 0 1 + β { 0 − 0} 0 β 1 dYt = dYt 1 dTt 1 + dYt dTt + dGt 1 + β { − − − } 1 + β { − } which is our key Intertemporal Keynesian Cross equation (23), where in this case

    m 0 dG0 mdT0   −  1 m m 0   dG (1 m) dT mdT   −   1 − − 0 − 1   ..   .  M =  0 1 m . 0  ∂Y =  .  (54)  −     .     . 0 m 0   dGt (1 m) dTt 1 mdTt     − − − −  . . . 0 . 1 m .. . − and m 1 is the static MPC. These equations come together with a restriction from assumption ≡ 1+β ?? that the government budget constraint is always satisﬁed, that is,

∑ dGt = ∑ dTt t 0 t 0 ≥ ≥ This infinite-dimensional linear system can therefore be written more compactly as (23). Lemmas 4 and ?? can then easily be verified. Any increase in government spending must result in an increase in current or future taxes. These taxes lower incomes, to which households respond by cutting consumption, to the point that the present-value effect must be nil. For example, consider two alternative schemes for dG0: one in which current taxes are raised, and one in which taxes are only raised after t periods. In the first case, if the government spends dG0 = 1 and raises taxes on the current generation, this in turn lowers spending by m today and by βm = 1 m tomorrow, −

50 hence G,tax h i0 ∂Y = 1 m (1 m) 0 dG0 (55) − − − ······ In the second case, we have

G,tax h i0 ∂Y = 1 0 m (1 m) 0 dG0 (56) · · · − − − ··· The government spends 1 at time 0, but the partial-equilibrium offset from private spending only occurs over the two periods during which the generation affected by the tax increase actually lives.

C Main proofs

C.1 Section3 proofs

C.1.1 Proof of proposition1

In an EYE given constant yt = y, inﬂation is constant at πt = 0 from (9). This, combined with the observation that yt = y, implies from both (Constant-r) or (Taylor rule), that the real interest rate only changes by drt = drt (respectively dit = drt). Given ﬂexible prices and no productivity shocks, real wages are constant at wt = 1. Given l = y, household hours are constant at their pt steady state level nit = Γ (eit, y).

The change in ﬁscal or monetary policy induces a change in government revenue of drevt =

(1 + rt) dbt 1 + bt 1drt + dgt dbt per period. We now see how this affects household net incomes − − − z = t + (1 τ ) e n . From (11) and (12) we have it t − t it it

ydτ = ydτr + (1 ϕ) drev t t − t and dt = ydτr ϕ drev t t − · t Hence net incomes are affected by

dz = dt e Γ (e , l) dτ it t − it it t e Γ (e , y) = ydτr ϕ drev it it (ydτr + (1 ϕ) drev ) t − · t − y t − t = ydτr (1 ω ) (ϕ + (1 ϕ) ω ) drev (57) t − i − − i t

eitΓ(eit,y) where ωi = y is household gross income relative to the mean. This delivers the proposition. r r Equation (57) shows that shocks to the tax rate dτt are just redistributive, with dτt > 0 increasing the incomes of agents below the mean and reducing those of agents above the mean. Meanwhile, any dollar change in E [dz ] = drev , so that any unit change in government revenue in a I it − t quarter ends up reducing household incomes by a unit on average, with ϕ determining whether this is paid mostly via a lump-sum or via a proportional reduction in incomes.

51 C.1.2 Proof of lemma3

According to our partial equibrium deﬁnition3, each household takes zit and maximizes (3) subject to

cit + ait = (1 + rt) ait 1 + zit t, i (4’) − ∀ t 1 Let qt ∏ = . Along any realized path, households satisfy ≡ s 0 1+rs ! ! T T (1 + r0) ai, 1 + ∑ qtzit = ∑ qtcit + q ai − t=0 t=0 T T

Taking expectations at date 0 for a given household, we have, ! ! T T (1 + r0) ai, 1 + ∑ qtE0[zit] = ∑ E0[cit] + q E0[ai ] − t=0 t=0 T T

Next, taking the population mean E of both sides gives, using iterated expectations E [E [ ]] = I I 0 · EI [ ], and noting that asset market clearing implies EI [ai, 1] = b 1 as well as E [ai,T] = b (the · − − T level of government debt outstanding), we have ! ! T T (1 + r0) b 1 + qtEI [zit] = qtEI [cit] + q b (58) − ∑ ∑ t=0 t=0 T T

But, from the deﬁnition of labor incomes and constant-y,

E [z ] = t + (1 τ ) E [e n ] I it t − t I it it = t + (1 τ ) y t − t = y rev − t = y gt + bt (1 + rt) bt 1 − − −

Now, telescoping the sum

T qt (bt (1 + rt) bt 1) = (1 + r0) b 1 + q b ∑ − − t=0 − − T T and therefore (58) is simply T ∑ qt (ct + gt y) = 0 t=0 − PE Hence, in our deﬁnition of an EYE ∂yt = dct + ∂gt, any shock must satisfy

T T ∑ qtd (ct + gt) + ∑ dqt (ct + gt y) = 0 t=0 t=0 × −

52 Hence T T ∑ qt∂yt + ∑ dqt 0 = 0 t=0 t=0 × where the ﬁrst term follows from our deﬁnition of ∂yt and the second term from goods market clearing at every date before the shock.

C.1.3 Proof of lemma4

[Analogous to lemma3, to be added.]

C.2 Section4 proofs

C.2.1 Proof of Theorem1

The proof of Theorem1 heavily relies on results and terminology from Markov chain potential theory, in particular the book by Kemeny et al.(1976), henceforth KSK. To use these results, we regard M as the transition matrix of a Markov chain and identify M with the associated Markov chain. Notice that by Assumption1, M is either recurrent or transient. First, suppose M is recurrent. Consider the direction where we are given a bounded regular vector v and that v ∑∞ Mk∂Y is finite-valued. Since ∂Y has zero NPV, this lets us apply 0 ≡ k=0 Theorem 9-15 in KSK to show that v0 is a particular solution to the IKC. Since v is regular, dY = v0 + v solves the IKC, too. Conversely, suppose dY is a finite-NPV solution to the IKC. Let λ = ∑∞ dY R be dY’s NPV. Then, if M is ergodic, this means that dY = ∑∞ Mk∂Y + v by Theorem t=0 t ∈ k=0 9-53, where v is unique regular vector with 10v = λ. If M is a null chain, this means that dY = ∞ k ∑k=0 M ∂Y by Corollary 9-17. ∞ k Second, suppose M is transient. In that case, ∂Y is always a charge and ∑k=0 M ∂Y is always a finite-valued solution to the IKC, see Theorem 8-3. This also means that any vector of the form ∞ k ∑k=0 M ∂Y + v where v is a regular vector solves the IKC. Conversely, if dY is a finite-valued solution, then (I M)(dY ∑∞ Mk∂Y) = 0 so dY and ∑∞ Mk∂Y differ precisely by a regular − − k=0 k=0 vector, implying (31).

C.2.2 Three helpful lemmas for determinacy

The following three lemmas turn out to be helpful for the proofs of our main results.

Lemma 7. Under Assumptions1 and2, the following holds:

a) if lim inft ∞ λ∗ > 0, there exists a λ > 0 such that lim sup pt(λ) < 1. → t

b) if lim supt ∞ λt∗ < 0, there exists a λ < 0 such that lim sup pt(λ) < 1. →

Proof. We prove part 1 only. Part 2 is analogous. Deﬁne λχ = χ lim inft ∞ λ∗ for all χ [0, 1]. By → t ∈ concavity, it follows that for all sufﬁciently large t, say t t, p (λ ) < 1. Suppose it were the case ≥ t 0.5

53 that p (λ ) 1 (or along some subsequence; the argument is analogous). Then, by concavity, t 0.5 → p (λ ) 1 for all χ. In fact, convergence (for large t) of χ p (λ ) is uniform, since each p (λ ) t χ → 7→ t χ t χ is bounded below by either by a line connecting point (0.5, pt(λ0.5)) with (0, 1) or with (1, pt(λ1)). λ Moreover, since pt(λ) interpreted as a polynomial in e is analytic, the limit is analytic too. But this means it must be that, point wise, M 0 unless s = t. A contradiction. s,t → Lemma 8. Let M be the (column-stochastic) transition matrix of a Markov chain with regular measure v = (vt), that is, Mv = v.

a) If f : N0 R+ is a strictly increasing function such that ∑s f (s)Ms,t > f (t) for all t outside a → ∞ ﬁnite set of states, then ∑t=0 f (t)vt = ∞.

b) If f : N0 R+ is a strictly decreasing function such that ∑s f (s)Ms,t < f (t) for all t outside a → ∞ ﬁnite set of states, then ∑t=0 f (t)vt = ∞.

Proof. We prove part 1. Part 2 is analogous. We are given that there exists a t 0 such that ∞ ≥ ∑ f (s)M , > f (t) for all t t. Assume by contradiction that ∑ f (t)v < ∞. In that case, we s s t ≥ t=0 t can split up the sum into two ﬁnite pieces,

∞ t ∞ ∑ f (t)vt = ∑ f (t)vt + ∑ f (t)vt, (59) t=0 t=0 t=t+1 which can both be bounded below.

For the ﬁrst sum, observe that vt = ∑s Mt,svs, thus

t t t ∞ t ∑ f (t)vt = ∑ ∑ f (t)Mt,svs + ∑ ∑ f (t)Mt,svs. t=0 s=0 t=0 s=t+1 t=0

Here, note that the second term is bounded above,

∞ t ∞ t ∑ ∑ f (t)Mt,svs < f (t) ∑ ∑ Mt,svs, (60) s=t+1 t=0 s=t+1 t=0 where we used the monotonicity of f . Further, by the regularity of v and the column-stochasticity of M we obtain,

∞ t t t ! t t t ∑ ∑ Mt,svs = ∑ vt ∑ Mt,svs = ∑ vt ∑ ∑ Mt,svs s=t+1 t=0 t=0 − s=0 t=0 − s=0 t=0

t t ∞ ! t ∞ = ∑ vt ∑ 1 ∑ Mt,s vs = ∑ ∑ Mt,svs, (61) t=0 − s=0 − t=t+1 s=0 t=t+1

54 so that, when substituting back into (60), we arrive at

∞ t t ∞ t ∞ ∑ ∑ f (t)Mt,svs < f (t) ∑ ∑ Mt,svs < ∑ ∑ f (t)Mt,svs. (62) s=t+1 t=0 s=0 t=t+1 s=0 t=t+1

Thus, using (62), the ﬁrst sum of (59) is bounded above by

t t t t ∞ t ∞ ∑ f (t)vt < ∑ ∑ f (t)Mt,svs + ∑ ∑ f (t)Mt,svs = ∑ ∑ f (t)Mt,svs. t=0 s=0 t=0 s=0 t=t+1 s=0 t=0

The second sum of (59) is also bounded above,

∞ ∞ ∞ ∑ f (t)vt < ∑ ∑ f (s)Ms,tvt t=t+1 t=t+1 s=0 where we used the fact that for large t, ∑s f (s)Ms,t > f (t). After relabeling of indices s and t, we then ﬁnd that

∞ t ∞ ∞ ∞ ∞ ∞ ∞ ∑ f (t)vt < ∑ ∑ f (t)Mt,svs + ∑ ∑ f (s)Ms,tvt = ∑ ∑ f (s)Ms,tvt = ∑ f (t)vt, t=0 s=0 t=0 t=t+1 s=0 t=0 s=0 t=0 which is a contradiction.

Lemma 9. Let M be the (column-stochastic) transition matrix of a recurrent Markov chain with unique

(up to scale) stationary measure v = (vt). If f : N0 R+ is a strictly increasing function such that there → ∞ exists e > 0 with ∑ f (s)M , < f (t)(1 e) for all t outside a ﬁnite set of states, then ∑ f (t)v < ∞. s s t − t=0 t Proof. This proof is analogous to the proof of Proposition 1.4 in Hairer(2016). Let t be such that

∑ f (s)M , < f (t)(1 e) for all t > t. Define f = f N for N > 0 and G by G (t) = s s t − N ∧ N N ∑ f (s)M , f (t). G is negative outside a finite set and satisfies G (t) ∑ f (s)M , f (t) t N s t − N N N → t s t − ≡ G(t) for all t (pointwise). By definition ∑t GN(t)vt = 0 and by Fatou’s lemma we therefore have

0 = lim inf GN(t)vt G(t)vt = G(t)vt + G(t)vt < G(t)vt e f (t)vt. ∞ ∑ ∑ ∑ ∑ ∑ ∑ N t ≤ t t t t>t t t − t>t → ≤ ≤ ∞ The sum ∑t=0 f (t)vt is then bounded above by

∞ 1 ∑ f (t)vt < e− ∑ G(t)vt + ∑ f (t)vt < ∞. t=0 t t t t ≤ ≤

55 C.2.3 Proof of Theorem2

Again we use the terminology in Kemeny et al.(1976). For part (a), distinguish two cases: M transient and M recurrent. If M is transient, Theorem XXX in Kemeny et al.(1976) proves that for any vector v with finite present value, 1 v < ∞, ∑∞ Mkv is finite-valued. This immedi- | 0 | k=0 ately implies that any right-eigenvector with eigenvalue 1 of M must have infinite present value 1 v , or the sum 1 v must be otherwise ill-defined. In particular, this rules out a case where v | 0 | 0 is bounded in current values since that would immediately imply summability of the elements of v. Thus, in that case, dY is locally determinate. If M is recurrent, let v be the unique (up to scale) stationary measure of M. Choose e > 0 so that r + e ( r, lim inf λ ) and define − ∈ − t∗ (r e)t f (t) = e − . By the strict convexity of pt(λ), it must then be that for sufficiently far out states ∞ t t for some t, ∑ f (s)M , > f (t). Applying Lemma8 below, we obtain that ∑ f (t)v = ∞, ≥ s s t t=0 t rt et so that e vt = e f (t)vt is necessarily unbounded (or else f (t)vt were summable). Therefore, dY is locally determinate.

Consider part (b). First notice that according to Lemma7, there exists a λ < 0 with lim supt ∞ pt(λ) < λs → 1. Second, M is necessarily recurrent, since 1 δ lim supt ∞ pt(λ) < 1 implies that ∑s e− Ms,t − ≡ → − e λt < δe λt < δ for sufﬁciently large t (and by negativity of λ). A straightforward application − − − − of Proposition 1.3 in Hairer(2016) establishes that M is recurrent.

Now, by convexity of pt(λ), the fact that lim supt ∞ λt∗ < r and lim supt ∞ pt(λ) < 1, it → − → follows that for any e (0, r), lim sup p ( r) = 1 δ < 1 for some other δ > 0. Defining ∈ t ∞ t − − rt → f (t) = e , this can be written as ∑ f (s)M , f (t) < δ f (t) for all sufficiently large t. Applying s t − − Lemma9 below, we then obtain that ∑∞ f (t)v < ∞ in this case, or, f (t)v 0. Therefore, v is t=0 t t → t bounded in current values and dY is locally indeterminate. The proof of the simple sufficient conditions is immediate.

C.2.4 Proof of Proposition3

By Lemma7, there exists a λ < 0 with lim sup p (λ) 1 δ < 1. t ∞ t ≡ − → λt Part (a): Deﬁne f : 0, 1, 2, . . . R+, f (t) e− . Then, lim supt ∞ pt(λ) < 1 implies that { } → ≡ → for all t sufﬁciently large, ∞ ∑ f (s)Ms,t < f (t) δ f (t). s=0 − By Proposition 1.3 in Hairer(2016), M is recurrent. In that case, the stationary measure of M is unique up to scale. The fact that any solution takes the form (33) is a direct consequence of the general IKC solution in Theorem1. Finally, an application of Theorem XXX in Kemeny et al.(1976) ∞ k proves that ∑k=0 M ∂Y has zero NPV.

Part (b): First, lim inft ∞ λt∗ > 0, together with lim supt ∞ pt(λ) < 1 and concavity of all pt, → → implies that λ (0, lim inft ∞ λ∗). Thus, for sufﬁciently large t, say t t, pt(λ) < 1 e, where ∈ → t ≥ −

56 e (0, 1 lim supt ∞ pt(λ)), meaning ∈ − → ∞ λ(s t) ∑ e− − Ms,t < 1 e, t t. (63) s=0 − ∀ ≥

Using Proposition 1.3 from Hairer(2016), this shows that M is transient. For that case, Kemeny ∞ k et al.(1976) prove that N ∑k=0 M is finite-valued (Theorem XXX), that Ns,t Ns,s for any ≡ 1 ≤ s, t 0 (Theorem XXX), and that Ns,s 1 Prs τ s < ∞ − (Theorem XXX). Here, Prs(A) ≥ ≤ − { } denotes the probability that the Markov chain associated with M moves from initial state s to set A 0, 1, 2, . . . ; and τ denotes the hitting time of set A once one period has passed, that is, ⊂ { } A if (X ) denotes the Markov chain, τ = min n 1 X A . We now prove that, under our n A { ≥ | n ∈ } assumptions, N is uniformly bounded (above) in t 0. Once that result is established, it follows t,t ≥ immediately that dY is bounded, entry by entry. Uniqueness follows from Theorem2. To prove that N is uniformly bounded above, we again define the function f : 0, 1, 2, . . . t,t { } → R , f (t) e λt. We establish the bound in three steps. First, pick two times t , t with t > t t + ≡ − 0 1 1 0 ≥ and define A 0, 1, . . . , t . Then,14 0 ≡ { 0} ∞ 1 1 1 λs f (t0)− f (t1) > f (t0)− e f (t1) + f (t0)− ∑ e− Ms,t s=0

t0 ∞ 1 λs 1 λs f (t )− e− M + f (t )− e− M ≥ 0 ∑ s,t 0 ∑ s,t s=0 s=t0+1 ∞ 1 Pr (A ) + f (t )− f (s)M , ≥ t1 0 ∑ 0 s,t s=t0+1 which by iteration, proves that f (t ) 1 f (t ) Pr (τ < ∞). Since f (t ) 1 f (t ) e λ 1 δ, 0 − 1 ≥ t1 A0 0 − 1 ≤ − ≡ − this proves that Pr (τ < ∞) < 1 δ t , t : t > t t. (64) t1 A0 − ∀ 0 1 1 0 ≥ Second, observe that (63) implies

∞ ∞ λ(s t) λ(s t) ∑ e− − 1 Ms,t < ∑ e− − 1 Ms,t < e s=t+1 − s=0 − − such that

∞ ∞ λ(s t) Prt ( t + 1, t + 2, . . . ) = ∑ Ms,t ∑ 1 e− − Ms,t > e t t. (65) { } s=t+1 ≥ s=t+1 − ∀ ≥

Finally, we bring conditions (64) and (65) together to establish uniform boundedness of Nt,t.

14This argument is analogous to one made in the proof of Proposition 1.3 in Hairer(2016).

57 Pick any t t. Then, 0 ≥

1 1 Nt−,t Prt0 τ t < ∞ = Prt0 ( t0 ) + Prt0 ( t1 )Prt1 (τ t < ∞) + Prt0 ( t1 ) Prt1 (τ t < ∞) − 0 0 ≤ { 0} { } ∑ { } { 0} ∑ { } { 0} t1t0 | {z } Pr (τ <∞) ≤ t1 A0 < 1 Pr ( t + 1, t + 2, . . . ) + Pr ( t + 1, t + 2, . . . ) (1 δ) − t0 { 0 0 } t0 { 0 0 } · − = 1 δ Pr ( t + 1, t + 2, . . . ) − · t0 { 0 0 } < 1 δe. −

Thus, N < (δe) 1 for t t, where e and δ are independent of t . This establishes that N is not t0,t0 − 0 ≥ 0 only ﬁnite-valued but also bounded, element-by-element.

C.2.5 Proof of Proposition5

We show this result in 2 steps: First, we show that any ordered matrix M preserves the “front- loaded” ordering under matrix multiplication. Second, we prove the main result.

Lemma 10. Let M RN N be an ordered, column-stochastic matrix. Let v RN be more front-loaded ∈ × 0 ∈ than v RN. Then, Mv is more front-loaded than Mv. ∈ 0 Proof.

Now we prove the main result. Notice that from Lemma 10 it immediately follows that ∞ k ∞ k ∑k=0 M ∂Y0 is more front-loaded than ∑k=0 M ∂Y. This establishes the result when M is transient (case (b) in Proposition3). When M is recurrent, w ∑∞ Mk∂Y and w ∑∞ Mk∂Y ≡ k=0 0 ≡ k=0 0 have zero NPV. In particular, this implies that eventually, for some t T for some T, w w . Let ≥ t0 ≥ t v 0 be the unique eigenvector of M with eigenvalue 1 , so we can write ∞ dY = ∑ Mk∂Y + λv, k=0 and similarly for dY . We have λ λ , and therefore dY is more front-loaded than dY. 0 ≥ 0 0

C.2.6 Proof of Theorem3

The idea behind the proof of Theorem3 is an application of Corollary2. In particular, we ﬁrst compute the asymptotic characteristic polynomial pjoint(λ) of the joint MPC matrix Mjoint and then check under what conditions pjoint( r) 1. − ≷ First, the joint MPC matrix is given by

t 1 joint R 1 − t t0 R M = Ms,t + M φy + κφπ + (φπ β− )κ (βR) − M , s,t s,t − ∑ s,t0 t0=0

58 with characteristic polynomial

∞ t 1 joint R 1 − λ(s t) (ρ r)(t t0) R p (λ) = pt(λ) + p (λ) φy + κφπ + (φπ β− )κ e− − e− − − M , t t − ∑ ∑ s,t0 s=0 t0=0 where we use the notation ρ log β. By exchanging the order of summation this simpliﬁes to ≡ − t 1 joint R 1 − (ρ r λ)(t t0) R p (λ) = pt(λ) + p (λ) φy + κφπ + (φπ β− )κ e− − − − p (λ). t t − ∑ t0 t0=0

Since M and MR were assumed to be self-similar, for large t and λ < ρ r this converges to − R joint R 1 p (λ) p (λ) = p(λ) + p (λ) φ + κφπ + (φπ β− )κ . y − (βR) 1e λ 1 − − − Rearranging, pjoint( r) 1 if and only if − ≷ 1 β 1 β 1 p( r) − φy + φπ ≷ 1 + − − − . κ κ pR( r) − The determinacy result then follows by applying Corollary2.

D Fiscal theory of the price level

To be added.